Lecture Notes For MATH305
Lecture Notes Lecture Notes 1-Notes on Fundamentals (Sections 1.1-1.6, 2.1) Extra Notes
Lecture Notes Lecture Notes 2-Notes on Analyticity (Sections 2.2--2.5) Proof of Theorem 2 on CR Equations (expanded version: page 75 of book)
Lecture Notes 2.5 Lecture Notes on Conformal Mappings and Laplace Equation
Lecture Notes 3 Lecture Notes 3- Notes on Some Simple Functions (Sections 3.1-3.2)
Lecture Notes 4 Lecture Notes 4-Notes on Multi-valued Functions (Sections 3.3 and 3.5) (Also take a look at these notes on branch cuts by Prof. Rosales of MIT) (Here are also a few additional carefully worked out problems with branch cuts)
Lecture Notes 4.5 Lecture Notes 4.5-Notes on inverse function of sin (z)
Lecture Notes 5 Lecture Notes 5-Notes on Contour Integration, Cauchy's Integral Theorem (Sections 4.1--4.3): First batch of notes Second batch of notes
Lecture Notes 6 Lecture Notes 6-Notes on Nyquist Criterion
Lecture Notes 6.5 Lecture Notes 6.5-Notes on Rouche's Theorem
Lecture Notes 7 Lecture Notes 7- Notes on Laurent Series, Singularities and Residue Calculus
Lecture Notes 8 Lecture Notes 8-Notes on Residue Calculus
Lecture Notes 9 Lecture Notes 9-Notes on Integration of Real Integrals by Residue Calculus
Lecture Notes 9-5 Lecture Notes 9-5: A Summary
Lecture Notes 10 Lecture Notes 10- Notes on Fourier Transforms and Applications
Downloads For MATH305
Download 1: Syllabus
Homework One (due: Jan.17) HW1
Homework Two (due: Jan.24) HW2
Homework Three (due: Jan.31) HW3
Homework Four (due: Feb. 7) HW3
Homework Five (due: Feb. 22) HW3
Outline of Teaching Plan For MATH 305
Jan. 10: Fundamentals of complex variable. Euler's formula. Polar coordinate. Principal value of argument Arg (z).
Jan. 12: Arg (z) and arg (z). De Moivre's formula. Roots of unit. Roots of a complex variable.
Jan. 14: Complex exponential. Sets in the complex plane. Functions of complex variables.
Jan. 17: Functions of complex variables. Image under linear and Mobius map $ w=(a+bz)/ (c+dz)$.
Jan. 19: Image under $w=z^2$. Continuous, differentiable, analytic. Cauchy-Riemann equation.
Jan. 21: Consequences of Cauchy-Riemann equation. Harmonic Functions. Conformal Mapping. Level Sets. More notes can be found here. Lecture Notes on Conformal Mappings
Jan. 24: Laplace under analytical mappings. $\partial_{\bar{z}} f(z)=0$. Conformal Mappings. Elementary Functions.
Jan. 26: Elementary functions $ e^z$ and $ \sin (z)$. Images under $ e^z$ and $ sin (z)$.
Jan. 28: Properties of $\sin z$ and $ sinh (z)$. Introduction of $Log (z)$.
Jan. 31: Multi-valued functions. introduction of $log (z)$ and $Log (z)$ and their properties.
Feb. 2: Multi-valued functions. Introduction of $z^\alpha$ and branch cuts.
Feb. 4: Multi-valued functions. Branch cuts for $ (z^2-1)^{\frac{1}{2}}$.
Feb. 7: Branch Cuts for $ (z^3-z)^{1/2}, (z^3-z)^{1/3}, (z^2+1)^{1/2}$.
Feb. 9: Inverse function of sin (z). Solving Laplace equation with Arg (z)
Feb. 11: Midterm 1
Feb. 14: Complex integrals. Contours (Paths).
Feb. 16: Fundamental Theorem of Calculus in the Complex Case. Examples.
Feb. 18: Cauchy-Coursat Theorem. simply-connected domains. Path independence and deformation of path.
Feb. 21-25: Spring Break
Feb. 28: Path Independence. Cauchy Integral Formula. Examples.
Mar. 2: Applications of Cauchy Integral Formula. Computation of real integrals.
Mar. 4: Consequences of Cauchy Integral Formula. Functions with finite order singularity.
Mar. 7: Consequences of Cauchy Integral Formula. Liouville Theorem: bounded entire functions are constants.
March 9: Maximum Modulus Principle.
March 11: Argument Principle,Rouche's Theorem (Lecture Note 6-5).
March 14: Nyquist criterion, applications to ODE.
March 16: Nyquist criterion. Introduction to Laurant series .(Lecture Note 7).
March 18:Introduction to Laurant series .(Lecture Note 7). Classification of singularities and computations of residues. See explanations of the notation O(...) here Notation $O(z^m)$
March 21: Lecture Note 7. Cauchy Residue Theorem. Computation of residues and contour integrals.
March 23: Applications of Cauchy Residue Theorem. Lecture Note 8, page 1-3.
March 25: Midterm 2
March 28: Type I, Type II real integrals.
March 30: Type III real integrals.
April 1: Type IV and Type V integrals. Integrals involving Multi-valued functions.
April 4: Type V integrals. Finish lecture 8.
April 6 (last class): Fourier transforms and inverse Fourier transforms. Two Properties. Applications to ODE and PDE.
APril 8: Review
Announcements For MATH 305
Back to My Home Page
Lecture Notes Lecture Notes 2-Notes on Analyticity (Sections 2.2--2.5) Proof of Theorem 2 on CR Equations (expanded version: page 75 of book)
Lecture Notes 2.5 Lecture Notes on Conformal Mappings and Laplace Equation
Lecture Notes 3 Lecture Notes 3- Notes on Some Simple Functions (Sections 3.1-3.2)
Lecture Notes 4 Lecture Notes 4-Notes on Multi-valued Functions (Sections 3.3 and 3.5) (Also take a look at these notes on branch cuts by Prof. Rosales of MIT) (Here are also a few additional carefully worked out problems with branch cuts)
Lecture Notes 4.5 Lecture Notes 4.5-Notes on inverse function of sin (z)
Lecture Notes 5 Lecture Notes 5-Notes on Contour Integration, Cauchy's Integral Theorem (Sections 4.1--4.3): First batch of notes Second batch of notes
Lecture Notes 6 Lecture Notes 6-Notes on Nyquist Criterion
Lecture Notes 6.5 Lecture Notes 6.5-Notes on Rouche's Theorem
Lecture Notes 7 Lecture Notes 7- Notes on Laurent Series, Singularities and Residue Calculus
Lecture Notes 8 Lecture Notes 8-Notes on Residue Calculus
Lecture Notes 9 Lecture Notes 9-Notes on Integration of Real Integrals by Residue Calculus
Lecture Notes 9-5 Lecture Notes 9-5: A Summary
Lecture Notes 10 Lecture Notes 10- Notes on Fourier Transforms and Applications
Downloads For MATH305
Download 1: Syllabus
Homework One (due: Jan.17) HW1
Homework Two (due: Jan.24) HW2
Homework Three (due: Jan.31) HW3
Homework Four (due: Feb. 7) HW3
Homework Five (due: Feb. 22) HW3
Outline of Teaching Plan For MATH 305
Jan. 10: Fundamentals of complex variable. Euler's formula. Polar coordinate. Principal value of argument Arg (z).
Jan. 12: Arg (z) and arg (z). De Moivre's formula. Roots of unit. Roots of a complex variable.
Jan. 14: Complex exponential. Sets in the complex plane. Functions of complex variables.
Jan. 17: Functions of complex variables. Image under linear and Mobius map $ w=(a+bz)/ (c+dz)$.
Jan. 19: Image under $w=z^2$. Continuous, differentiable, analytic. Cauchy-Riemann equation.
Jan. 21: Consequences of Cauchy-Riemann equation. Harmonic Functions. Conformal Mapping. Level Sets. More notes can be found here. Lecture Notes on Conformal Mappings
Jan. 24: Laplace under analytical mappings. $\partial_{\bar{z}} f(z)=0$. Conformal Mappings. Elementary Functions.
Jan. 26: Elementary functions $ e^z$ and $ \sin (z)$. Images under $ e^z$ and $ sin (z)$.
Jan. 28: Properties of $\sin z$ and $ sinh (z)$. Introduction of $Log (z)$.
Jan. 31: Multi-valued functions. introduction of $log (z)$ and $Log (z)$ and their properties.
Feb. 2: Multi-valued functions. Introduction of $z^\alpha$ and branch cuts.
Feb. 4: Multi-valued functions. Branch cuts for $ (z^2-1)^{\frac{1}{2}}$.
Feb. 7: Branch Cuts for $ (z^3-z)^{1/2}, (z^3-z)^{1/3}, (z^2+1)^{1/2}$.
Feb. 9: Inverse function of sin (z). Solving Laplace equation with Arg (z)
Feb. 11: Midterm 1
Feb. 14: Complex integrals. Contours (Paths).
Feb. 16: Fundamental Theorem of Calculus in the Complex Case. Examples.
Feb. 18: Cauchy-Coursat Theorem. simply-connected domains. Path independence and deformation of path.
Feb. 21-25: Spring Break
Feb. 28: Path Independence. Cauchy Integral Formula. Examples.
Mar. 2: Applications of Cauchy Integral Formula. Computation of real integrals.
Mar. 4: Consequences of Cauchy Integral Formula. Functions with finite order singularity.
Mar. 7: Consequences of Cauchy Integral Formula. Liouville Theorem: bounded entire functions are constants.
March 9: Maximum Modulus Principle.
March 11: Argument Principle,Rouche's Theorem (Lecture Note 6-5).
March 14: Nyquist criterion, applications to ODE.
March 16: Nyquist criterion. Introduction to Laurant series .(Lecture Note 7).
March 18:Introduction to Laurant series .(Lecture Note 7). Classification of singularities and computations of residues. See explanations of the notation O(...) here Notation $O(z^m)$
March 21: Lecture Note 7. Cauchy Residue Theorem. Computation of residues and contour integrals.
March 23: Applications of Cauchy Residue Theorem. Lecture Note 8, page 1-3.
March 25: Midterm 2
March 28: Type I, Type II real integrals.
March 30: Type III real integrals.
April 1: Type IV and Type V integrals. Integrals involving Multi-valued functions.
April 4: Type V integrals. Finish lecture 8.
April 6 (last class): Fourier transforms and inverse Fourier transforms. Two Properties. Applications to ODE and PDE.
APril 8: Review
Announcements For MATH 305
Back to My Home Page
Lecture Notes 2.5 Lecture Notes on Conformal Mappings and Laplace Equation
Lecture Notes 3 Lecture Notes 3- Notes on Some Simple Functions (Sections 3.1-3.2)
Lecture Notes 4 Lecture Notes 4-Notes on Multi-valued Functions (Sections 3.3 and 3.5) (Also take a look at these notes on branch cuts by Prof. Rosales of MIT) (Here are also a few additional carefully worked out problems with branch cuts)
Lecture Notes 4.5 Lecture Notes 4.5-Notes on inverse function of sin (z)
Lecture Notes 5 Lecture Notes 5-Notes on Contour Integration, Cauchy's Integral Theorem (Sections 4.1--4.3): First batch of notes Second batch of notes
Lecture Notes 6 Lecture Notes 6-Notes on Nyquist Criterion
Lecture Notes 6.5 Lecture Notes 6.5-Notes on Rouche's Theorem
Lecture Notes 7 Lecture Notes 7- Notes on Laurent Series, Singularities and Residue Calculus
Lecture Notes 8 Lecture Notes 8-Notes on Residue Calculus
Lecture Notes 9 Lecture Notes 9-Notes on Integration of Real Integrals by Residue Calculus
Lecture Notes 9-5 Lecture Notes 9-5: A Summary
Lecture Notes 10 Lecture Notes 10- Notes on Fourier Transforms and Applications
Downloads For MATH305
Download 1: Syllabus
Homework One (due: Jan.17) HW1
Homework Two (due: Jan.24) HW2
Homework Three (due: Jan.31) HW3
Homework Four (due: Feb. 7) HW3
Homework Five (due: Feb. 22) HW3
Outline of Teaching Plan For MATH 305
Jan. 10: Fundamentals of complex variable. Euler's formula. Polar coordinate. Principal value of argument Arg (z).
Jan. 12: Arg (z) and arg (z). De Moivre's formula. Roots of unit. Roots of a complex variable.
Jan. 14: Complex exponential. Sets in the complex plane. Functions of complex variables.
Jan. 17: Functions of complex variables. Image under linear and Mobius map $ w=(a+bz)/ (c+dz)$.
Jan. 19: Image under $w=z^2$. Continuous, differentiable, analytic. Cauchy-Riemann equation.
Jan. 21: Consequences of Cauchy-Riemann equation. Harmonic Functions. Conformal Mapping. Level Sets. More notes can be found here. Lecture Notes on Conformal Mappings
Jan. 24: Laplace under analytical mappings. $\partial_{\bar{z}} f(z)=0$. Conformal Mappings. Elementary Functions.
Jan. 26: Elementary functions $ e^z$ and $ \sin (z)$. Images under $ e^z$ and $ sin (z)$.
Jan. 28: Properties of $\sin z$ and $ sinh (z)$. Introduction of $Log (z)$.
Jan. 31: Multi-valued functions. introduction of $log (z)$ and $Log (z)$ and their properties.
Feb. 2: Multi-valued functions. Introduction of $z^\alpha$ and branch cuts.
Feb. 4: Multi-valued functions. Branch cuts for $ (z^2-1)^{\frac{1}{2}}$.
Feb. 7: Branch Cuts for $ (z^3-z)^{1/2}, (z^3-z)^{1/3}, (z^2+1)^{1/2}$.
Feb. 9: Inverse function of sin (z). Solving Laplace equation with Arg (z)
Feb. 11: Midterm 1
Feb. 14: Complex integrals. Contours (Paths).
Feb. 16: Fundamental Theorem of Calculus in the Complex Case. Examples.
Feb. 18: Cauchy-Coursat Theorem. simply-connected domains. Path independence and deformation of path.
Feb. 21-25: Spring Break
Feb. 28: Path Independence. Cauchy Integral Formula. Examples.
Mar. 2: Applications of Cauchy Integral Formula. Computation of real integrals.
Mar. 4: Consequences of Cauchy Integral Formula. Functions with finite order singularity.
Mar. 7: Consequences of Cauchy Integral Formula. Liouville Theorem: bounded entire functions are constants.
March 9: Maximum Modulus Principle.
March 11: Argument Principle,Rouche's Theorem (Lecture Note 6-5).
March 14: Nyquist criterion, applications to ODE.
March 16: Nyquist criterion. Introduction to Laurant series .(Lecture Note 7).
March 18:Introduction to Laurant series .(Lecture Note 7). Classification of singularities and computations of residues. See explanations of the notation O(...) here Notation $O(z^m)$
March 21: Lecture Note 7. Cauchy Residue Theorem. Computation of residues and contour integrals.
March 23: Applications of Cauchy Residue Theorem. Lecture Note 8, page 1-3.
March 25: Midterm 2
March 28: Type I, Type II real integrals.
March 30: Type III real integrals.
April 1: Type IV and Type V integrals. Integrals involving Multi-valued functions.
April 4: Type V integrals. Finish lecture 8.
April 6 (last class): Fourier transforms and inverse Fourier transforms. Two Properties. Applications to ODE and PDE.
APril 8: Review
Announcements For MATH 305
Back to My Home Page
Lecture Notes 3 Lecture Notes 3- Notes on Some Simple Functions (Sections 3.1-3.2)
Lecture Notes 4 Lecture Notes 4-Notes on Multi-valued Functions (Sections 3.3 and 3.5) (Also take a look at these notes on branch cuts by Prof. Rosales of MIT) (Here are also a few additional carefully worked out problems with branch cuts)
Lecture Notes 4.5 Lecture Notes 4.5-Notes on inverse function of sin (z)
Lecture Notes 5 Lecture Notes 5-Notes on Contour Integration, Cauchy's Integral Theorem (Sections 4.1--4.3): First batch of notes Second batch of notes
Lecture Notes 6 Lecture Notes 6-Notes on Nyquist Criterion
Lecture Notes 6.5 Lecture Notes 6.5-Notes on Rouche's Theorem
Lecture Notes 7 Lecture Notes 7- Notes on Laurent Series, Singularities and Residue Calculus
Lecture Notes 8 Lecture Notes 8-Notes on Residue Calculus
Lecture Notes 9 Lecture Notes 9-Notes on Integration of Real Integrals by Residue Calculus
Lecture Notes 9-5 Lecture Notes 9-5: A Summary
Lecture Notes 10 Lecture Notes 10- Notes on Fourier Transforms and Applications
Downloads For MATH305
Download 1: Syllabus
Homework One (due: Jan.17) HW1
Homework Two (due: Jan.24) HW2
Homework Three (due: Jan.31) HW3
Homework Four (due: Feb. 7) HW3
Homework Five (due: Feb. 22) HW3
Outline of Teaching Plan For MATH 305
Jan. 10: Fundamentals of complex variable. Euler's formula. Polar coordinate. Principal value of argument Arg (z).
Jan. 12: Arg (z) and arg (z). De Moivre's formula. Roots of unit. Roots of a complex variable.
Jan. 14: Complex exponential. Sets in the complex plane. Functions of complex variables.
Jan. 17: Functions of complex variables. Image under linear and Mobius map $ w=(a+bz)/ (c+dz)$.
Jan. 19: Image under $w=z^2$. Continuous, differentiable, analytic. Cauchy-Riemann equation.
Jan. 21: Consequences of Cauchy-Riemann equation. Harmonic Functions. Conformal Mapping. Level Sets. More notes can be found here. Lecture Notes on Conformal Mappings
Jan. 24: Laplace under analytical mappings. $\partial_{\bar{z}} f(z)=0$. Conformal Mappings. Elementary Functions.
Jan. 26: Elementary functions $ e^z$ and $ \sin (z)$. Images under $ e^z$ and $ sin (z)$.
Jan. 28: Properties of $\sin z$ and $ sinh (z)$. Introduction of $Log (z)$.
Jan. 31: Multi-valued functions. introduction of $log (z)$ and $Log (z)$ and their properties.
Feb. 2: Multi-valued functions. Introduction of $z^\alpha$ and branch cuts.
Feb. 4: Multi-valued functions. Branch cuts for $ (z^2-1)^{\frac{1}{2}}$.
Feb. 7: Branch Cuts for $ (z^3-z)^{1/2}, (z^3-z)^{1/3}, (z^2+1)^{1/2}$.
Feb. 9: Inverse function of sin (z). Solving Laplace equation with Arg (z)
Feb. 11: Midterm 1
Feb. 14: Complex integrals. Contours (Paths).
Feb. 16: Fundamental Theorem of Calculus in the Complex Case. Examples.
Feb. 18: Cauchy-Coursat Theorem. simply-connected domains. Path independence and deformation of path.
Feb. 21-25: Spring Break
Feb. 28: Path Independence. Cauchy Integral Formula. Examples.
Mar. 2: Applications of Cauchy Integral Formula. Computation of real integrals.
Mar. 4: Consequences of Cauchy Integral Formula. Functions with finite order singularity.
Mar. 7: Consequences of Cauchy Integral Formula. Liouville Theorem: bounded entire functions are constants.
March 9: Maximum Modulus Principle.
March 11: Argument Principle,Rouche's Theorem (Lecture Note 6-5).
March 14: Nyquist criterion, applications to ODE.
March 16: Nyquist criterion. Introduction to Laurant series .(Lecture Note 7).
March 18:Introduction to Laurant series .(Lecture Note 7). Classification of singularities and computations of residues. See explanations of the notation O(...) here Notation $O(z^m)$
March 21: Lecture Note 7. Cauchy Residue Theorem. Computation of residues and contour integrals.
March 23: Applications of Cauchy Residue Theorem. Lecture Note 8, page 1-3.
March 25: Midterm 2
March 28: Type I, Type II real integrals.
March 30: Type III real integrals.
April 1: Type IV and Type V integrals. Integrals involving Multi-valued functions.
April 4: Type V integrals. Finish lecture 8.
April 6 (last class): Fourier transforms and inverse Fourier transforms. Two Properties. Applications to ODE and PDE.
APril 8: Review
Announcements For MATH 305
Back to My Home Page
Lecture Notes 4 Lecture Notes 4-Notes on Multi-valued Functions (Sections 3.3 and 3.5) (Also take a look at these notes on branch cuts by Prof. Rosales of MIT) (Here are also a few additional carefully worked out problems with branch cuts)
Lecture Notes 4.5 Lecture Notes 4.5-Notes on inverse function of sin (z)
Lecture Notes 5 Lecture Notes 5-Notes on Contour Integration, Cauchy's Integral Theorem (Sections 4.1--4.3): First batch of notes Second batch of notes
Lecture Notes 6 Lecture Notes 6-Notes on Nyquist Criterion
Lecture Notes 6.5 Lecture Notes 6.5-Notes on Rouche's Theorem
Lecture Notes 7 Lecture Notes 7- Notes on Laurent Series, Singularities and Residue Calculus
Lecture Notes 8 Lecture Notes 8-Notes on Residue Calculus
Lecture Notes 9 Lecture Notes 9-Notes on Integration of Real Integrals by Residue Calculus
Lecture Notes 9-5 Lecture Notes 9-5: A Summary
Lecture Notes 10 Lecture Notes 10- Notes on Fourier Transforms and Applications
Downloads For MATH305
Download 1: Syllabus
Homework One (due: Jan.17) HW1
Homework Two (due: Jan.24) HW2
Homework Three (due: Jan.31) HW3
Homework Four (due: Feb. 7) HW3
Homework Five (due: Feb. 22) HW3
Outline of Teaching Plan For MATH 305
Jan. 10: Fundamentals of complex variable. Euler's formula. Polar coordinate. Principal value of argument Arg (z).
Jan. 12: Arg (z) and arg (z). De Moivre's formula. Roots of unit. Roots of a complex variable.
Jan. 14: Complex exponential. Sets in the complex plane. Functions of complex variables.
Jan. 17: Functions of complex variables. Image under linear and Mobius map $ w=(a+bz)/ (c+dz)$.
Jan. 19: Image under $w=z^2$. Continuous, differentiable, analytic. Cauchy-Riemann equation.
Jan. 21: Consequences of Cauchy-Riemann equation. Harmonic Functions. Conformal Mapping. Level Sets. More notes can be found here. Lecture Notes on Conformal Mappings
Jan. 24: Laplace under analytical mappings. $\partial_{\bar{z}} f(z)=0$. Conformal Mappings. Elementary Functions.
Jan. 26: Elementary functions $ e^z$ and $ \sin (z)$. Images under $ e^z$ and $ sin (z)$.
Jan. 28: Properties of $\sin z$ and $ sinh (z)$. Introduction of $Log (z)$.
Jan. 31: Multi-valued functions. introduction of $log (z)$ and $Log (z)$ and their properties.
Feb. 2: Multi-valued functions. Introduction of $z^\alpha$ and branch cuts.
Feb. 4: Multi-valued functions. Branch cuts for $ (z^2-1)^{\frac{1}{2}}$.
Feb. 7: Branch Cuts for $ (z^3-z)^{1/2}, (z^3-z)^{1/3}, (z^2+1)^{1/2}$.
Feb. 9: Inverse function of sin (z). Solving Laplace equation with Arg (z)
Feb. 11: Midterm 1
Feb. 14: Complex integrals. Contours (Paths).
Feb. 16: Fundamental Theorem of Calculus in the Complex Case. Examples.
Feb. 18: Cauchy-Coursat Theorem. simply-connected domains. Path independence and deformation of path.
Feb. 21-25: Spring Break
Feb. 28: Path Independence. Cauchy Integral Formula. Examples.
Mar. 2: Applications of Cauchy Integral Formula. Computation of real integrals.
Mar. 4: Consequences of Cauchy Integral Formula. Functions with finite order singularity.
Mar. 7: Consequences of Cauchy Integral Formula. Liouville Theorem: bounded entire functions are constants.
March 9: Maximum Modulus Principle.
March 11: Argument Principle,Rouche's Theorem (Lecture Note 6-5).
March 14: Nyquist criterion, applications to ODE.
March 16: Nyquist criterion. Introduction to Laurant series .(Lecture Note 7).
March 18:Introduction to Laurant series .(Lecture Note 7). Classification of singularities and computations of residues. See explanations of the notation O(...) here Notation $O(z^m)$
March 21: Lecture Note 7. Cauchy Residue Theorem. Computation of residues and contour integrals.
March 23: Applications of Cauchy Residue Theorem. Lecture Note 8, page 1-3.
March 25: Midterm 2
March 28: Type I, Type II real integrals.
March 30: Type III real integrals.
April 1: Type IV and Type V integrals. Integrals involving Multi-valued functions.
April 4: Type V integrals. Finish lecture 8.
April 6 (last class): Fourier transforms and inverse Fourier transforms. Two Properties. Applications to ODE and PDE.
APril 8: Review
Announcements For MATH 305
Back to My Home Page
Lecture Notes 4.5 Lecture Notes 4.5-Notes on inverse function of sin (z)
Lecture Notes 5 Lecture Notes 5-Notes on Contour Integration, Cauchy's Integral Theorem (Sections 4.1--4.3): First batch of notes Second batch of notes
Lecture Notes 6 Lecture Notes 6-Notes on Nyquist Criterion
Lecture Notes 6.5 Lecture Notes 6.5-Notes on Rouche's Theorem
Lecture Notes 7 Lecture Notes 7- Notes on Laurent Series, Singularities and Residue Calculus
Lecture Notes 8 Lecture Notes 8-Notes on Residue Calculus
Lecture Notes 9 Lecture Notes 9-Notes on Integration of Real Integrals by Residue Calculus
Lecture Notes 9-5 Lecture Notes 9-5: A Summary
Lecture Notes 10 Lecture Notes 10- Notes on Fourier Transforms and Applications
Downloads For MATH305
Download 1: Syllabus
Homework One (due: Jan.17) HW1
Homework Two (due: Jan.24) HW2
Homework Three (due: Jan.31) HW3
Homework Four (due: Feb. 7) HW3
Homework Five (due: Feb. 22) HW3
Outline of Teaching Plan For MATH 305
Jan. 10: Fundamentals of complex variable. Euler's formula. Polar coordinate. Principal value of argument Arg (z).
Jan. 12: Arg (z) and arg (z). De Moivre's formula. Roots of unit. Roots of a complex variable.
Jan. 14: Complex exponential. Sets in the complex plane. Functions of complex variables.
Jan. 17: Functions of complex variables. Image under linear and Mobius map $ w=(a+bz)/ (c+dz)$.
Jan. 19: Image under $w=z^2$. Continuous, differentiable, analytic. Cauchy-Riemann equation.
Jan. 21: Consequences of Cauchy-Riemann equation. Harmonic Functions. Conformal Mapping. Level Sets. More notes can be found here. Lecture Notes on Conformal Mappings
Jan. 24: Laplace under analytical mappings. $\partial_{\bar{z}} f(z)=0$. Conformal Mappings. Elementary Functions.
Jan. 26: Elementary functions $ e^z$ and $ \sin (z)$. Images under $ e^z$ and $ sin (z)$.
Jan. 28: Properties of $\sin z$ and $ sinh (z)$. Introduction of $Log (z)$.
Jan. 31: Multi-valued functions. introduction of $log (z)$ and $Log (z)$ and their properties.
Feb. 2: Multi-valued functions. Introduction of $z^\alpha$ and branch cuts.
Feb. 4: Multi-valued functions. Branch cuts for $ (z^2-1)^{\frac{1}{2}}$.
Feb. 7: Branch Cuts for $ (z^3-z)^{1/2}, (z^3-z)^{1/3}, (z^2+1)^{1/2}$.
Feb. 9: Inverse function of sin (z). Solving Laplace equation with Arg (z)
Feb. 11: Midterm 1
Feb. 14: Complex integrals. Contours (Paths).
Feb. 16: Fundamental Theorem of Calculus in the Complex Case. Examples.
Feb. 18: Cauchy-Coursat Theorem. simply-connected domains. Path independence and deformation of path.
Feb. 21-25: Spring Break
Feb. 28: Path Independence. Cauchy Integral Formula. Examples.
Mar. 2: Applications of Cauchy Integral Formula. Computation of real integrals.
Mar. 4: Consequences of Cauchy Integral Formula. Functions with finite order singularity.
Mar. 7: Consequences of Cauchy Integral Formula. Liouville Theorem: bounded entire functions are constants.
March 9: Maximum Modulus Principle.
March 11: Argument Principle,Rouche's Theorem (Lecture Note 6-5).
March 14: Nyquist criterion, applications to ODE.
March 16: Nyquist criterion. Introduction to Laurant series .(Lecture Note 7).
March 18:Introduction to Laurant series .(Lecture Note 7). Classification of singularities and computations of residues. See explanations of the notation O(...) here Notation $O(z^m)$
March 21: Lecture Note 7. Cauchy Residue Theorem. Computation of residues and contour integrals.
March 23: Applications of Cauchy Residue Theorem. Lecture Note 8, page 1-3.
March 25: Midterm 2
March 28: Type I, Type II real integrals.
March 30: Type III real integrals.
April 1: Type IV and Type V integrals. Integrals involving Multi-valued functions.
April 4: Type V integrals. Finish lecture 8.
April 6 (last class): Fourier transforms and inverse Fourier transforms. Two Properties. Applications to ODE and PDE.
APril 8: Review
Announcements For MATH 305
Back to My Home Page
Lecture Notes 5 Lecture Notes 5-Notes on Contour Integration, Cauchy's Integral Theorem (Sections 4.1--4.3): First batch of notes Second batch of notes
Lecture Notes 6 Lecture Notes 6-Notes on Nyquist Criterion
Lecture Notes 6.5 Lecture Notes 6.5-Notes on Rouche's Theorem
Lecture Notes 7 Lecture Notes 7- Notes on Laurent Series, Singularities and Residue Calculus
Lecture Notes 8 Lecture Notes 8-Notes on Residue Calculus
Lecture Notes 9 Lecture Notes 9-Notes on Integration of Real Integrals by Residue Calculus
Lecture Notes 9-5 Lecture Notes 9-5: A Summary
Lecture Notes 10 Lecture Notes 10- Notes on Fourier Transforms and Applications
Downloads For MATH305
Download 1: Syllabus
Homework One (due: Jan.17) HW1
Homework Two (due: Jan.24) HW2
Homework Three (due: Jan.31) HW3
Homework Four (due: Feb. 7) HW3
Homework Five (due: Feb. 22) HW3
Outline of Teaching Plan For MATH 305
Jan. 10: Fundamentals of complex variable. Euler's formula. Polar coordinate. Principal value of argument Arg (z).
Jan. 12: Arg (z) and arg (z). De Moivre's formula. Roots of unit. Roots of a complex variable.
Jan. 14: Complex exponential. Sets in the complex plane. Functions of complex variables.
Jan. 17: Functions of complex variables. Image under linear and Mobius map $ w=(a+bz)/ (c+dz)$.
Jan. 19: Image under $w=z^2$. Continuous, differentiable, analytic. Cauchy-Riemann equation.
Jan. 21: Consequences of Cauchy-Riemann equation. Harmonic Functions. Conformal Mapping. Level Sets. More notes can be found here. Lecture Notes on Conformal Mappings
Jan. 24: Laplace under analytical mappings. $\partial_{\bar{z}} f(z)=0$. Conformal Mappings. Elementary Functions.
Jan. 26: Elementary functions $ e^z$ and $ \sin (z)$. Images under $ e^z$ and $ sin (z)$.
Jan. 28: Properties of $\sin z$ and $ sinh (z)$. Introduction of $Log (z)$.
Jan. 31: Multi-valued functions. introduction of $log (z)$ and $Log (z)$ and their properties.
Feb. 2: Multi-valued functions. Introduction of $z^\alpha$ and branch cuts.
Feb. 4: Multi-valued functions. Branch cuts for $ (z^2-1)^{\frac{1}{2}}$.
Feb. 7: Branch Cuts for $ (z^3-z)^{1/2}, (z^3-z)^{1/3}, (z^2+1)^{1/2}$.
Feb. 9: Inverse function of sin (z). Solving Laplace equation with Arg (z)
Feb. 11: Midterm 1
Feb. 14: Complex integrals. Contours (Paths).
Feb. 16: Fundamental Theorem of Calculus in the Complex Case. Examples.
Feb. 18: Cauchy-Coursat Theorem. simply-connected domains. Path independence and deformation of path.
Feb. 21-25: Spring Break
Feb. 28: Path Independence. Cauchy Integral Formula. Examples.
Mar. 2: Applications of Cauchy Integral Formula. Computation of real integrals.
Mar. 4: Consequences of Cauchy Integral Formula. Functions with finite order singularity.
Mar. 7: Consequences of Cauchy Integral Formula. Liouville Theorem: bounded entire functions are constants.
March 9: Maximum Modulus Principle.
March 11: Argument Principle,Rouche's Theorem (Lecture Note 6-5).
March 14: Nyquist criterion, applications to ODE.
March 16: Nyquist criterion. Introduction to Laurant series .(Lecture Note 7).
March 18:Introduction to Laurant series .(Lecture Note 7). Classification of singularities and computations of residues. See explanations of the notation O(...) here Notation $O(z^m)$
March 21: Lecture Note 7. Cauchy Residue Theorem. Computation of residues and contour integrals.
March 23: Applications of Cauchy Residue Theorem. Lecture Note 8, page 1-3.
March 25: Midterm 2
March 28: Type I, Type II real integrals.
March 30: Type III real integrals.
April 1: Type IV and Type V integrals. Integrals involving Multi-valued functions.
April 4: Type V integrals. Finish lecture 8.
April 6 (last class): Fourier transforms and inverse Fourier transforms. Two Properties. Applications to ODE and PDE.
APril 8: Review
Announcements For MATH 305
Back to My Home Page
Lecture Notes 6 Lecture Notes 6-Notes on Nyquist Criterion
Lecture Notes 6.5 Lecture Notes 6.5-Notes on Rouche's Theorem
Lecture Notes 7 Lecture Notes 7- Notes on Laurent Series, Singularities and Residue Calculus
Lecture Notes 8 Lecture Notes 8-Notes on Residue Calculus
Lecture Notes 9 Lecture Notes 9-Notes on Integration of Real Integrals by Residue Calculus
Lecture Notes 9-5 Lecture Notes 9-5: A Summary
Lecture Notes 10 Lecture Notes 10- Notes on Fourier Transforms and Applications
Downloads For MATH305
Download 1: Syllabus
Homework One (due: Jan.17) HW1
Homework Two (due: Jan.24) HW2
Homework Three (due: Jan.31) HW3
Homework Four (due: Feb. 7) HW3
Homework Five (due: Feb. 22) HW3
Outline of Teaching Plan For MATH 305
Jan. 10: Fundamentals of complex variable. Euler's formula. Polar coordinate. Principal value of argument Arg (z).
Jan. 12: Arg (z) and arg (z). De Moivre's formula. Roots of unit. Roots of a complex variable.
Jan. 14: Complex exponential. Sets in the complex plane. Functions of complex variables.
Jan. 17: Functions of complex variables. Image under linear and Mobius map $ w=(a+bz)/ (c+dz)$.
Jan. 19: Image under $w=z^2$. Continuous, differentiable, analytic. Cauchy-Riemann equation.
Jan. 21: Consequences of Cauchy-Riemann equation. Harmonic Functions. Conformal Mapping. Level Sets. More notes can be found here. Lecture Notes on Conformal Mappings
Jan. 24: Laplace under analytical mappings. $\partial_{\bar{z}} f(z)=0$. Conformal Mappings. Elementary Functions.
Jan. 26: Elementary functions $ e^z$ and $ \sin (z)$. Images under $ e^z$ and $ sin (z)$.
Jan. 28: Properties of $\sin z$ and $ sinh (z)$. Introduction of $Log (z)$.
Jan. 31: Multi-valued functions. introduction of $log (z)$ and $Log (z)$ and their properties.
Feb. 2: Multi-valued functions. Introduction of $z^\alpha$ and branch cuts.
Feb. 4: Multi-valued functions. Branch cuts for $ (z^2-1)^{\frac{1}{2}}$.
Feb. 7: Branch Cuts for $ (z^3-z)^{1/2}, (z^3-z)^{1/3}, (z^2+1)^{1/2}$.
Feb. 9: Inverse function of sin (z). Solving Laplace equation with Arg (z)
Feb. 11: Midterm 1
Feb. 14: Complex integrals. Contours (Paths).
Feb. 16: Fundamental Theorem of Calculus in the Complex Case. Examples.
Feb. 18: Cauchy-Coursat Theorem. simply-connected domains. Path independence and deformation of path.
Feb. 21-25: Spring Break
Feb. 28: Path Independence. Cauchy Integral Formula. Examples.
Mar. 2: Applications of Cauchy Integral Formula. Computation of real integrals.
Mar. 4: Consequences of Cauchy Integral Formula. Functions with finite order singularity.
Mar. 7: Consequences of Cauchy Integral Formula. Liouville Theorem: bounded entire functions are constants.
March 9: Maximum Modulus Principle.
March 11: Argument Principle,Rouche's Theorem (Lecture Note 6-5).
March 14: Nyquist criterion, applications to ODE.
March 16: Nyquist criterion. Introduction to Laurant series .(Lecture Note 7).
March 18:Introduction to Laurant series .(Lecture Note 7). Classification of singularities and computations of residues. See explanations of the notation O(...) here Notation $O(z^m)$
March 21: Lecture Note 7. Cauchy Residue Theorem. Computation of residues and contour integrals.
March 23: Applications of Cauchy Residue Theorem. Lecture Note 8, page 1-3.
March 25: Midterm 2
March 28: Type I, Type II real integrals.
March 30: Type III real integrals.
April 1: Type IV and Type V integrals. Integrals involving Multi-valued functions.
April 4: Type V integrals. Finish lecture 8.
April 6 (last class): Fourier transforms and inverse Fourier transforms. Two Properties. Applications to ODE and PDE.
APril 8: Review
Announcements For MATH 305
Back to My Home Page
Lecture Notes 6.5 Lecture Notes 6.5-Notes on Rouche's Theorem
Lecture Notes 7 Lecture Notes 7- Notes on Laurent Series, Singularities and Residue Calculus
Lecture Notes 8 Lecture Notes 8-Notes on Residue Calculus
Lecture Notes 9 Lecture Notes 9-Notes on Integration of Real Integrals by Residue Calculus
Lecture Notes 9-5 Lecture Notes 9-5: A Summary
Lecture Notes 10 Lecture Notes 10- Notes on Fourier Transforms and Applications
Downloads For MATH305
Download 1: Syllabus
Homework One (due: Jan.17) HW1
Homework Two (due: Jan.24) HW2
Homework Three (due: Jan.31) HW3
Homework Four (due: Feb. 7) HW3
Homework Five (due: Feb. 22) HW3
Outline of Teaching Plan For MATH 305
Jan. 10: Fundamentals of complex variable. Euler's formula. Polar coordinate. Principal value of argument Arg (z).
Jan. 12: Arg (z) and arg (z). De Moivre's formula. Roots of unit. Roots of a complex variable.
Jan. 14: Complex exponential. Sets in the complex plane. Functions of complex variables.
Jan. 17: Functions of complex variables. Image under linear and Mobius map $ w=(a+bz)/ (c+dz)$.
Jan. 19: Image under $w=z^2$. Continuous, differentiable, analytic. Cauchy-Riemann equation.
Jan. 21: Consequences of Cauchy-Riemann equation. Harmonic Functions. Conformal Mapping. Level Sets. More notes can be found here. Lecture Notes on Conformal Mappings
Jan. 24: Laplace under analytical mappings. $\partial_{\bar{z}} f(z)=0$. Conformal Mappings. Elementary Functions.
Jan. 26: Elementary functions $ e^z$ and $ \sin (z)$. Images under $ e^z$ and $ sin (z)$.
Jan. 28: Properties of $\sin z$ and $ sinh (z)$. Introduction of $Log (z)$.
Jan. 31: Multi-valued functions. introduction of $log (z)$ and $Log (z)$ and their properties.
Feb. 2: Multi-valued functions. Introduction of $z^\alpha$ and branch cuts.
Feb. 4: Multi-valued functions. Branch cuts for $ (z^2-1)^{\frac{1}{2}}$.
Feb. 7: Branch Cuts for $ (z^3-z)^{1/2}, (z^3-z)^{1/3}, (z^2+1)^{1/2}$.
Feb. 9: Inverse function of sin (z). Solving Laplace equation with Arg (z)
Feb. 11: Midterm 1
Feb. 14: Complex integrals. Contours (Paths).
Feb. 16: Fundamental Theorem of Calculus in the Complex Case. Examples.
Feb. 18: Cauchy-Coursat Theorem. simply-connected domains. Path independence and deformation of path.
Feb. 21-25: Spring Break
Feb. 28: Path Independence. Cauchy Integral Formula. Examples.
Mar. 2: Applications of Cauchy Integral Formula. Computation of real integrals.
Mar. 4: Consequences of Cauchy Integral Formula. Functions with finite order singularity.
Mar. 7: Consequences of Cauchy Integral Formula. Liouville Theorem: bounded entire functions are constants.
March 9: Maximum Modulus Principle.
March 11: Argument Principle,Rouche's Theorem (Lecture Note 6-5).
March 14: Nyquist criterion, applications to ODE.
March 16: Nyquist criterion. Introduction to Laurant series .(Lecture Note 7).
March 18:Introduction to Laurant series .(Lecture Note 7). Classification of singularities and computations of residues. See explanations of the notation O(...) here Notation $O(z^m)$
March 21: Lecture Note 7. Cauchy Residue Theorem. Computation of residues and contour integrals.
March 23: Applications of Cauchy Residue Theorem. Lecture Note 8, page 1-3.
March 25: Midterm 2
March 28: Type I, Type II real integrals.
March 30: Type III real integrals.
April 1: Type IV and Type V integrals. Integrals involving Multi-valued functions.
April 4: Type V integrals. Finish lecture 8.
April 6 (last class): Fourier transforms and inverse Fourier transforms. Two Properties. Applications to ODE and PDE.
APril 8: Review
Announcements For MATH 305
Back to My Home Page
Lecture Notes 7 Lecture Notes 7- Notes on Laurent Series, Singularities and Residue Calculus
Lecture Notes 8 Lecture Notes 8-Notes on Residue Calculus
Lecture Notes 9 Lecture Notes 9-Notes on Integration of Real Integrals by Residue Calculus
Lecture Notes 9-5 Lecture Notes 9-5: A Summary
Lecture Notes 10 Lecture Notes 10- Notes on Fourier Transforms and Applications
Downloads For MATH305
Download 1: Syllabus
Homework One (due: Jan.17) HW1
Homework Two (due: Jan.24) HW2
Homework Three (due: Jan.31) HW3
Homework Four (due: Feb. 7) HW3
Homework Five (due: Feb. 22) HW3
Outline of Teaching Plan For MATH 305
Jan. 10: Fundamentals of complex variable. Euler's formula. Polar coordinate. Principal value of argument Arg (z).
Jan. 12: Arg (z) and arg (z). De Moivre's formula. Roots of unit. Roots of a complex variable.
Jan. 14: Complex exponential. Sets in the complex plane. Functions of complex variables.
Jan. 17: Functions of complex variables. Image under linear and Mobius map $ w=(a+bz)/ (c+dz)$.
Jan. 19: Image under $w=z^2$. Continuous, differentiable, analytic. Cauchy-Riemann equation.
Jan. 21: Consequences of Cauchy-Riemann equation. Harmonic Functions. Conformal Mapping. Level Sets. More notes can be found here. Lecture Notes on Conformal Mappings
Jan. 24: Laplace under analytical mappings. $\partial_{\bar{z}} f(z)=0$. Conformal Mappings. Elementary Functions.
Jan. 26: Elementary functions $ e^z$ and $ \sin (z)$. Images under $ e^z$ and $ sin (z)$.
Jan. 28: Properties of $\sin z$ and $ sinh (z)$. Introduction of $Log (z)$.
Jan. 31: Multi-valued functions. introduction of $log (z)$ and $Log (z)$ and their properties.
Feb. 2: Multi-valued functions. Introduction of $z^\alpha$ and branch cuts.
Feb. 4: Multi-valued functions. Branch cuts for $ (z^2-1)^{\frac{1}{2}}$.
Feb. 7: Branch Cuts for $ (z^3-z)^{1/2}, (z^3-z)^{1/3}, (z^2+1)^{1/2}$.
Feb. 9: Inverse function of sin (z). Solving Laplace equation with Arg (z)
Feb. 11: Midterm 1
Feb. 14: Complex integrals. Contours (Paths).
Feb. 16: Fundamental Theorem of Calculus in the Complex Case. Examples.
Feb. 18: Cauchy-Coursat Theorem. simply-connected domains. Path independence and deformation of path.
Feb. 21-25: Spring Break
Feb. 28: Path Independence. Cauchy Integral Formula. Examples.
Mar. 2: Applications of Cauchy Integral Formula. Computation of real integrals.
Mar. 4: Consequences of Cauchy Integral Formula. Functions with finite order singularity.
Mar. 7: Consequences of Cauchy Integral Formula. Liouville Theorem: bounded entire functions are constants.
March 9: Maximum Modulus Principle.
March 11: Argument Principle,Rouche's Theorem (Lecture Note 6-5).
March 14: Nyquist criterion, applications to ODE.
March 16: Nyquist criterion. Introduction to Laurant series .(Lecture Note 7).
March 18:Introduction to Laurant series .(Lecture Note 7). Classification of singularities and computations of residues. See explanations of the notation O(...) here Notation $O(z^m)$
March 21: Lecture Note 7. Cauchy Residue Theorem. Computation of residues and contour integrals.
March 23: Applications of Cauchy Residue Theorem. Lecture Note 8, page 1-3.
March 25: Midterm 2
March 28: Type I, Type II real integrals.
March 30: Type III real integrals.
April 1: Type IV and Type V integrals. Integrals involving Multi-valued functions.
April 4: Type V integrals. Finish lecture 8.
April 6 (last class): Fourier transforms and inverse Fourier transforms. Two Properties. Applications to ODE and PDE.
APril 8: Review
Announcements For MATH 305
Back to My Home Page
Lecture Notes 8 Lecture Notes 8-Notes on Residue Calculus
Lecture Notes 9 Lecture Notes 9-Notes on Integration of Real Integrals by Residue Calculus
Lecture Notes 9-5 Lecture Notes 9-5: A Summary
Lecture Notes 10 Lecture Notes 10- Notes on Fourier Transforms and Applications
Downloads For MATH305
Download 1: Syllabus
Homework One (due: Jan.17) HW1
Homework Two (due: Jan.24) HW2
Homework Three (due: Jan.31) HW3
Homework Four (due: Feb. 7) HW3
Homework Five (due: Feb. 22) HW3
Outline of Teaching Plan For MATH 305
Jan. 10: Fundamentals of complex variable. Euler's formula. Polar coordinate. Principal value of argument Arg (z).
Jan. 12: Arg (z) and arg (z). De Moivre's formula. Roots of unit. Roots of a complex variable.
Jan. 14: Complex exponential. Sets in the complex plane. Functions of complex variables.
Jan. 17: Functions of complex variables. Image under linear and Mobius map $ w=(a+bz)/ (c+dz)$.
Jan. 19: Image under $w=z^2$. Continuous, differentiable, analytic. Cauchy-Riemann equation.
Jan. 21: Consequences of Cauchy-Riemann equation. Harmonic Functions. Conformal Mapping. Level Sets. More notes can be found here. Lecture Notes on Conformal Mappings
Jan. 24: Laplace under analytical mappings. $\partial_{\bar{z}} f(z)=0$. Conformal Mappings. Elementary Functions.
Jan. 26: Elementary functions $ e^z$ and $ \sin (z)$. Images under $ e^z$ and $ sin (z)$.
Jan. 28: Properties of $\sin z$ and $ sinh (z)$. Introduction of $Log (z)$.
Jan. 31: Multi-valued functions. introduction of $log (z)$ and $Log (z)$ and their properties.
Feb. 2: Multi-valued functions. Introduction of $z^\alpha$ and branch cuts.
Feb. 4: Multi-valued functions. Branch cuts for $ (z^2-1)^{\frac{1}{2}}$.
Feb. 7: Branch Cuts for $ (z^3-z)^{1/2}, (z^3-z)^{1/3}, (z^2+1)^{1/2}$.
Feb. 9: Inverse function of sin (z). Solving Laplace equation with Arg (z)
Feb. 11: Midterm 1
Feb. 14: Complex integrals. Contours (Paths).
Feb. 16: Fundamental Theorem of Calculus in the Complex Case. Examples.
Feb. 18: Cauchy-Coursat Theorem. simply-connected domains. Path independence and deformation of path.
Feb. 21-25: Spring Break
Feb. 28: Path Independence. Cauchy Integral Formula. Examples.
Mar. 2: Applications of Cauchy Integral Formula. Computation of real integrals.
Mar. 4: Consequences of Cauchy Integral Formula. Functions with finite order singularity.
Mar. 7: Consequences of Cauchy Integral Formula. Liouville Theorem: bounded entire functions are constants.
March 9: Maximum Modulus Principle.
March 11: Argument Principle,Rouche's Theorem (Lecture Note 6-5).
March 14: Nyquist criterion, applications to ODE.
March 16: Nyquist criterion. Introduction to Laurant series .(Lecture Note 7).
March 18:Introduction to Laurant series .(Lecture Note 7). Classification of singularities and computations of residues. See explanations of the notation O(...) here Notation $O(z^m)$
March 21: Lecture Note 7. Cauchy Residue Theorem. Computation of residues and contour integrals.
March 23: Applications of Cauchy Residue Theorem. Lecture Note 8, page 1-3.
March 25: Midterm 2
March 28: Type I, Type II real integrals.
March 30: Type III real integrals.
April 1: Type IV and Type V integrals. Integrals involving Multi-valued functions.
April 4: Type V integrals. Finish lecture 8.
April 6 (last class): Fourier transforms and inverse Fourier transforms. Two Properties. Applications to ODE and PDE.
APril 8: Review
Announcements For MATH 305
Back to My Home Page
Lecture Notes 9 Lecture Notes 9-Notes on Integration of Real Integrals by Residue Calculus
Lecture Notes 9-5 Lecture Notes 9-5: A Summary
Lecture Notes 10 Lecture Notes 10- Notes on Fourier Transforms and Applications
Downloads For MATH305
Download 1: Syllabus
Homework One (due: Jan.17) HW1
Homework Two (due: Jan.24) HW2
Homework Three (due: Jan.31) HW3
Homework Four (due: Feb. 7) HW3
Homework Five (due: Feb. 22) HW3
Outline of Teaching Plan For MATH 305
Jan. 10: Fundamentals of complex variable. Euler's formula. Polar coordinate. Principal value of argument Arg (z).
Jan. 12: Arg (z) and arg (z). De Moivre's formula. Roots of unit. Roots of a complex variable.
Jan. 14: Complex exponential. Sets in the complex plane. Functions of complex variables.
Jan. 17: Functions of complex variables. Image under linear and Mobius map $ w=(a+bz)/ (c+dz)$.
Jan. 19: Image under $w=z^2$. Continuous, differentiable, analytic. Cauchy-Riemann equation.
Jan. 21: Consequences of Cauchy-Riemann equation. Harmonic Functions. Conformal Mapping. Level Sets. More notes can be found here. Lecture Notes on Conformal Mappings
Jan. 24: Laplace under analytical mappings. $\partial_{\bar{z}} f(z)=0$. Conformal Mappings. Elementary Functions.
Jan. 26: Elementary functions $ e^z$ and $ \sin (z)$. Images under $ e^z$ and $ sin (z)$.
Jan. 28: Properties of $\sin z$ and $ sinh (z)$. Introduction of $Log (z)$.
Jan. 31: Multi-valued functions. introduction of $log (z)$ and $Log (z)$ and their properties.
Feb. 2: Multi-valued functions. Introduction of $z^\alpha$ and branch cuts.
Feb. 4: Multi-valued functions. Branch cuts for $ (z^2-1)^{\frac{1}{2}}$.
Feb. 7: Branch Cuts for $ (z^3-z)^{1/2}, (z^3-z)^{1/3}, (z^2+1)^{1/2}$.
Feb. 9: Inverse function of sin (z). Solving Laplace equation with Arg (z)
Feb. 11: Midterm 1
Feb. 14: Complex integrals. Contours (Paths).
Feb. 16: Fundamental Theorem of Calculus in the Complex Case. Examples.
Feb. 18: Cauchy-Coursat Theorem. simply-connected domains. Path independence and deformation of path.
Feb. 21-25: Spring Break
Feb. 28: Path Independence. Cauchy Integral Formula. Examples.
Mar. 2: Applications of Cauchy Integral Formula. Computation of real integrals.
Mar. 4: Consequences of Cauchy Integral Formula. Functions with finite order singularity.
Mar. 7: Consequences of Cauchy Integral Formula. Liouville Theorem: bounded entire functions are constants.
March 9: Maximum Modulus Principle.
March 11: Argument Principle,Rouche's Theorem (Lecture Note 6-5).
March 14: Nyquist criterion, applications to ODE.
March 16: Nyquist criterion. Introduction to Laurant series .(Lecture Note 7).
March 18:Introduction to Laurant series .(Lecture Note 7). Classification of singularities and computations of residues. See explanations of the notation O(...) here Notation $O(z^m)$
March 21: Lecture Note 7. Cauchy Residue Theorem. Computation of residues and contour integrals.
March 23: Applications of Cauchy Residue Theorem. Lecture Note 8, page 1-3.
March 25: Midterm 2
March 28: Type I, Type II real integrals.
March 30: Type III real integrals.
April 1: Type IV and Type V integrals. Integrals involving Multi-valued functions.
April 4: Type V integrals. Finish lecture 8.
April 6 (last class): Fourier transforms and inverse Fourier transforms. Two Properties. Applications to ODE and PDE.
APril 8: Review
Announcements For MATH 305
Back to My Home Page
Lecture Notes 9-5 Lecture Notes 9-5: A Summary
Lecture Notes 10 Lecture Notes 10- Notes on Fourier Transforms and Applications
Downloads For MATH305
Download 1: Syllabus
Homework One (due: Jan.17) HW1
Homework Two (due: Jan.24) HW2
Homework Three (due: Jan.31) HW3
Homework Four (due: Feb. 7) HW3
Homework Five (due: Feb. 22) HW3
Outline of Teaching Plan For MATH 305
Jan. 10: Fundamentals of complex variable. Euler's formula. Polar coordinate. Principal value of argument Arg (z).
Jan. 12: Arg (z) and arg (z). De Moivre's formula. Roots of unit. Roots of a complex variable.
Jan. 14: Complex exponential. Sets in the complex plane. Functions of complex variables.
Jan. 17: Functions of complex variables. Image under linear and Mobius map $ w=(a+bz)/ (c+dz)$.
Jan. 19: Image under $w=z^2$. Continuous, differentiable, analytic. Cauchy-Riemann equation.
Jan. 21: Consequences of Cauchy-Riemann equation. Harmonic Functions. Conformal Mapping. Level Sets. More notes can be found here. Lecture Notes on Conformal Mappings
Jan. 24: Laplace under analytical mappings. $\partial_{\bar{z}} f(z)=0$. Conformal Mappings. Elementary Functions.
Jan. 26: Elementary functions $ e^z$ and $ \sin (z)$. Images under $ e^z$ and $ sin (z)$.
Jan. 28: Properties of $\sin z$ and $ sinh (z)$. Introduction of $Log (z)$.
Jan. 31: Multi-valued functions. introduction of $log (z)$ and $Log (z)$ and their properties.
Feb. 2: Multi-valued functions. Introduction of $z^\alpha$ and branch cuts.
Feb. 4: Multi-valued functions. Branch cuts for $ (z^2-1)^{\frac{1}{2}}$.
Feb. 7: Branch Cuts for $ (z^3-z)^{1/2}, (z^3-z)^{1/3}, (z^2+1)^{1/2}$.
Feb. 9: Inverse function of sin (z). Solving Laplace equation with Arg (z)
Feb. 11: Midterm 1
Feb. 14: Complex integrals. Contours (Paths).
Feb. 16: Fundamental Theorem of Calculus in the Complex Case. Examples.
Feb. 18: Cauchy-Coursat Theorem. simply-connected domains. Path independence and deformation of path.
Feb. 21-25: Spring Break
Feb. 28: Path Independence. Cauchy Integral Formula. Examples.
Mar. 2: Applications of Cauchy Integral Formula. Computation of real integrals.
Mar. 4: Consequences of Cauchy Integral Formula. Functions with finite order singularity.
Mar. 7: Consequences of Cauchy Integral Formula. Liouville Theorem: bounded entire functions are constants.
March 9: Maximum Modulus Principle.
March 11: Argument Principle,Rouche's Theorem (Lecture Note 6-5).
March 14: Nyquist criterion, applications to ODE.
March 16: Nyquist criterion. Introduction to Laurant series .(Lecture Note 7).
March 18:Introduction to Laurant series .(Lecture Note 7). Classification of singularities and computations of residues. See explanations of the notation O(...) here Notation $O(z^m)$
March 21: Lecture Note 7. Cauchy Residue Theorem. Computation of residues and contour integrals.
March 23: Applications of Cauchy Residue Theorem. Lecture Note 8, page 1-3.
March 25: Midterm 2
March 28: Type I, Type II real integrals.
March 30: Type III real integrals.
April 1: Type IV and Type V integrals. Integrals involving Multi-valued functions.
April 4: Type V integrals. Finish lecture 8.
April 6 (last class): Fourier transforms and inverse Fourier transforms. Two Properties. Applications to ODE and PDE.
APril 8: Review
Announcements For MATH 305
Back to My Home Page
Lecture Notes 10 Lecture Notes 10- Notes on Fourier Transforms and Applications
Downloads For MATH305
Download 1: Syllabus
Homework One (due: Jan.17) HW1
Homework Two (due: Jan.24) HW2
Homework Three (due: Jan.31) HW3
Homework Four (due: Feb. 7) HW3
Homework Five (due: Feb. 22) HW3
Outline of Teaching Plan For MATH 305
Jan. 10: Fundamentals of complex variable. Euler's formula. Polar coordinate. Principal value of argument Arg (z).
Jan. 12: Arg (z) and arg (z). De Moivre's formula. Roots of unit. Roots of a complex variable.
Jan. 14: Complex exponential. Sets in the complex plane. Functions of complex variables.
Jan. 17: Functions of complex variables. Image under linear and Mobius map $ w=(a+bz)/ (c+dz)$.
Jan. 19: Image under $w=z^2$. Continuous, differentiable, analytic. Cauchy-Riemann equation.
Jan. 21: Consequences of Cauchy-Riemann equation. Harmonic Functions. Conformal Mapping. Level Sets. More notes can be found here. Lecture Notes on Conformal Mappings
Jan. 24: Laplace under analytical mappings. $\partial_{\bar{z}} f(z)=0$. Conformal Mappings. Elementary Functions.
Jan. 26: Elementary functions $ e^z$ and $ \sin (z)$. Images under $ e^z$ and $ sin (z)$.
Jan. 28: Properties of $\sin z$ and $ sinh (z)$. Introduction of $Log (z)$.
Jan. 31: Multi-valued functions. introduction of $log (z)$ and $Log (z)$ and their properties.
Feb. 2: Multi-valued functions. Introduction of $z^\alpha$ and branch cuts.
Feb. 4: Multi-valued functions. Branch cuts for $ (z^2-1)^{\frac{1}{2}}$.
Feb. 7: Branch Cuts for $ (z^3-z)^{1/2}, (z^3-z)^{1/3}, (z^2+1)^{1/2}$.
Feb. 9: Inverse function of sin (z). Solving Laplace equation with Arg (z)
Feb. 11: Midterm 1
Feb. 14: Complex integrals. Contours (Paths).
Feb. 16: Fundamental Theorem of Calculus in the Complex Case. Examples.
Feb. 18: Cauchy-Coursat Theorem. simply-connected domains. Path independence and deformation of path.
Feb. 21-25: Spring Break
Feb. 28: Path Independence. Cauchy Integral Formula. Examples.
Mar. 2: Applications of Cauchy Integral Formula. Computation of real integrals.
Mar. 4: Consequences of Cauchy Integral Formula. Functions with finite order singularity.
Mar. 7: Consequences of Cauchy Integral Formula. Liouville Theorem: bounded entire functions are constants.
March 9: Maximum Modulus Principle.
March 11: Argument Principle,Rouche's Theorem (Lecture Note 6-5).
March 14: Nyquist criterion, applications to ODE.
March 16: Nyquist criterion. Introduction to Laurant series .(Lecture Note 7).
March 18:Introduction to Laurant series .(Lecture Note 7). Classification of singularities and computations of residues. See explanations of the notation O(...) here Notation $O(z^m)$
March 21: Lecture Note 7. Cauchy Residue Theorem. Computation of residues and contour integrals.
March 23: Applications of Cauchy Residue Theorem. Lecture Note 8, page 1-3.
March 25: Midterm 2
March 28: Type I, Type II real integrals.
March 30: Type III real integrals.
April 1: Type IV and Type V integrals. Integrals involving Multi-valued functions.
April 4: Type V integrals. Finish lecture 8.
April 6 (last class): Fourier transforms and inverse Fourier transforms. Two Properties. Applications to ODE and PDE.
APril 8: Review
Announcements For MATH 305
Back to My Home Page
Downloads For MATH305
Download 1: Syllabus
Homework One (due: Jan.17) HW1
Homework Two (due: Jan.24) HW2
Homework Three (due: Jan.31) HW3
Homework Four (due: Feb. 7) HW3
Homework Five (due: Feb. 22) HW3
Outline of Teaching Plan For MATH 305
Jan. 10: Fundamentals of complex variable. Euler's formula. Polar coordinate. Principal value of argument Arg (z).
Jan. 12: Arg (z) and arg (z). De Moivre's formula. Roots of unit. Roots of a complex variable.
Jan. 14: Complex exponential. Sets in the complex plane. Functions of complex variables.
Jan. 17: Functions of complex variables. Image under linear and Mobius map $ w=(a+bz)/ (c+dz)$.
Jan. 19: Image under $w=z^2$. Continuous, differentiable, analytic. Cauchy-Riemann equation.
Jan. 21: Consequences of Cauchy-Riemann equation. Harmonic Functions. Conformal Mapping. Level Sets. More notes can be found here. Lecture Notes on Conformal Mappings
Jan. 24: Laplace under analytical mappings. $\partial_{\bar{z}} f(z)=0$. Conformal Mappings. Elementary Functions.
Jan. 26: Elementary functions $ e^z$ and $ \sin (z)$. Images under $ e^z$ and $ sin (z)$.
Jan. 28: Properties of $\sin z$ and $ sinh (z)$. Introduction of $Log (z)$.
Jan. 31: Multi-valued functions. introduction of $log (z)$ and $Log (z)$ and their properties.
Feb. 2: Multi-valued functions. Introduction of $z^\alpha$ and branch cuts.
Feb. 4: Multi-valued functions. Branch cuts for $ (z^2-1)^{\frac{1}{2}}$.
Feb. 7: Branch Cuts for $ (z^3-z)^{1/2}, (z^3-z)^{1/3}, (z^2+1)^{1/2}$.
Feb. 9: Inverse function of sin (z). Solving Laplace equation with Arg (z)
Feb. 11: Midterm 1
Feb. 14: Complex integrals. Contours (Paths).
Feb. 16: Fundamental Theorem of Calculus in the Complex Case. Examples.
Feb. 18: Cauchy-Coursat Theorem. simply-connected domains. Path independence and deformation of path.
Feb. 21-25: Spring Break
Feb. 28: Path Independence. Cauchy Integral Formula. Examples.
Mar. 2: Applications of Cauchy Integral Formula. Computation of real integrals.
Mar. 4: Consequences of Cauchy Integral Formula. Functions with finite order singularity.
Mar. 7: Consequences of Cauchy Integral Formula. Liouville Theorem: bounded entire functions are constants.
March 9: Maximum Modulus Principle.
March 11: Argument Principle,Rouche's Theorem (Lecture Note 6-5).
March 14: Nyquist criterion, applications to ODE.
March 16: Nyquist criterion. Introduction to Laurant series .(Lecture Note 7).
March 18:Introduction to Laurant series .(Lecture Note 7). Classification of singularities and computations of residues. See explanations of the notation O(...) here Notation $O(z^m)$
March 21: Lecture Note 7. Cauchy Residue Theorem. Computation of residues and contour integrals.
March 23: Applications of Cauchy Residue Theorem. Lecture Note 8, page 1-3.
March 25: Midterm 2
March 28: Type I, Type II real integrals.
March 30: Type III real integrals.
April 1: Type IV and Type V integrals. Integrals involving Multi-valued functions.
April 4: Type V integrals. Finish lecture 8.
April 6 (last class): Fourier transforms and inverse Fourier transforms. Two Properties. Applications to ODE and PDE.
APril 8: Review
Announcements For MATH 305
Back to My Home Page
Homework Two (due: Jan.24) HW2
Homework Three (due: Jan.31) HW3
Homework Four (due: Feb. 7) HW3
Homework Five (due: Feb. 22) HW3
Outline of Teaching Plan For MATH 305
Jan. 10: Fundamentals of complex variable. Euler's formula. Polar coordinate. Principal value of argument Arg (z).
Jan. 12: Arg (z) and arg (z). De Moivre's formula. Roots of unit. Roots of a complex variable.
Jan. 14: Complex exponential. Sets in the complex plane. Functions of complex variables.
Jan. 17: Functions of complex variables. Image under linear and Mobius map $ w=(a+bz)/ (c+dz)$.
Jan. 19: Image under $w=z^2$. Continuous, differentiable, analytic. Cauchy-Riemann equation.
Jan. 21: Consequences of Cauchy-Riemann equation. Harmonic Functions. Conformal Mapping. Level Sets. More notes can be found here. Lecture Notes on Conformal Mappings
Jan. 24: Laplace under analytical mappings. $\partial_{\bar{z}} f(z)=0$. Conformal Mappings. Elementary Functions.
Jan. 26: Elementary functions $ e^z$ and $ \sin (z)$. Images under $ e^z$ and $ sin (z)$.
Jan. 28: Properties of $\sin z$ and $ sinh (z)$. Introduction of $Log (z)$.
Jan. 31: Multi-valued functions. introduction of $log (z)$ and $Log (z)$ and their properties.
Feb. 2: Multi-valued functions. Introduction of $z^\alpha$ and branch cuts.
Feb. 4: Multi-valued functions. Branch cuts for $ (z^2-1)^{\frac{1}{2}}$.
Feb. 7: Branch Cuts for $ (z^3-z)^{1/2}, (z^3-z)^{1/3}, (z^2+1)^{1/2}$.
Feb. 9: Inverse function of sin (z). Solving Laplace equation with Arg (z)
Feb. 11: Midterm 1
Feb. 14: Complex integrals. Contours (Paths).
Feb. 16: Fundamental Theorem of Calculus in the Complex Case. Examples.
Feb. 18: Cauchy-Coursat Theorem. simply-connected domains. Path independence and deformation of path.
Feb. 21-25: Spring Break
Feb. 28: Path Independence. Cauchy Integral Formula. Examples.
Mar. 2: Applications of Cauchy Integral Formula. Computation of real integrals.
Mar. 4: Consequences of Cauchy Integral Formula. Functions with finite order singularity.
Mar. 7: Consequences of Cauchy Integral Formula. Liouville Theorem: bounded entire functions are constants.
March 9: Maximum Modulus Principle.
March 11: Argument Principle,Rouche's Theorem (Lecture Note 6-5).
March 14: Nyquist criterion, applications to ODE.
March 16: Nyquist criterion. Introduction to Laurant series .(Lecture Note 7).
March 18:Introduction to Laurant series .(Lecture Note 7). Classification of singularities and computations of residues. See explanations of the notation O(...) here Notation $O(z^m)$
March 21: Lecture Note 7. Cauchy Residue Theorem. Computation of residues and contour integrals.
March 23: Applications of Cauchy Residue Theorem. Lecture Note 8, page 1-3.
March 25: Midterm 2
March 28: Type I, Type II real integrals.
March 30: Type III real integrals.
April 1: Type IV and Type V integrals. Integrals involving Multi-valued functions.
April 4: Type V integrals. Finish lecture 8.
April 6 (last class): Fourier transforms and inverse Fourier transforms. Two Properties. Applications to ODE and PDE.
APril 8: Review
Announcements For MATH 305
Back to My Home Page
Homework Three (due: Jan.31) HW3
Homework Four (due: Feb. 7) HW3
Homework Five (due: Feb. 22) HW3
Outline of Teaching Plan For MATH 305
Jan. 10: Fundamentals of complex variable. Euler's formula. Polar coordinate. Principal value of argument Arg (z).
Jan. 12: Arg (z) and arg (z). De Moivre's formula. Roots of unit. Roots of a complex variable.
Jan. 14: Complex exponential. Sets in the complex plane. Functions of complex variables.
Jan. 17: Functions of complex variables. Image under linear and Mobius map $ w=(a+bz)/ (c+dz)$.
Jan. 19: Image under $w=z^2$. Continuous, differentiable, analytic. Cauchy-Riemann equation.
Jan. 21: Consequences of Cauchy-Riemann equation. Harmonic Functions. Conformal Mapping. Level Sets. More notes can be found here. Lecture Notes on Conformal Mappings
Jan. 24: Laplace under analytical mappings. $\partial_{\bar{z}} f(z)=0$. Conformal Mappings. Elementary Functions.
Jan. 26: Elementary functions $ e^z$ and $ \sin (z)$. Images under $ e^z$ and $ sin (z)$.
Jan. 28: Properties of $\sin z$ and $ sinh (z)$. Introduction of $Log (z)$.
Jan. 31: Multi-valued functions. introduction of $log (z)$ and $Log (z)$ and their properties.
Feb. 2: Multi-valued functions. Introduction of $z^\alpha$ and branch cuts.
Feb. 4: Multi-valued functions. Branch cuts for $ (z^2-1)^{\frac{1}{2}}$.
Feb. 7: Branch Cuts for $ (z^3-z)^{1/2}, (z^3-z)^{1/3}, (z^2+1)^{1/2}$.
Feb. 9: Inverse function of sin (z). Solving Laplace equation with Arg (z)
Feb. 11: Midterm 1
Feb. 14: Complex integrals. Contours (Paths).
Feb. 16: Fundamental Theorem of Calculus in the Complex Case. Examples.
Feb. 18: Cauchy-Coursat Theorem. simply-connected domains. Path independence and deformation of path.
Feb. 21-25: Spring Break
Feb. 28: Path Independence. Cauchy Integral Formula. Examples.
Mar. 2: Applications of Cauchy Integral Formula. Computation of real integrals.
Mar. 4: Consequences of Cauchy Integral Formula. Functions with finite order singularity.
Mar. 7: Consequences of Cauchy Integral Formula. Liouville Theorem: bounded entire functions are constants.
March 9: Maximum Modulus Principle.
March 11: Argument Principle,Rouche's Theorem (Lecture Note 6-5).
March 14: Nyquist criterion, applications to ODE.
March 16: Nyquist criterion. Introduction to Laurant series .(Lecture Note 7).
March 18:Introduction to Laurant series .(Lecture Note 7). Classification of singularities and computations of residues. See explanations of the notation O(...) here Notation $O(z^m)$
March 21: Lecture Note 7. Cauchy Residue Theorem. Computation of residues and contour integrals.
March 23: Applications of Cauchy Residue Theorem. Lecture Note 8, page 1-3.
March 25: Midterm 2
March 28: Type I, Type II real integrals.
March 30: Type III real integrals.
April 1: Type IV and Type V integrals. Integrals involving Multi-valued functions.
April 4: Type V integrals. Finish lecture 8.
April 6 (last class): Fourier transforms and inverse Fourier transforms. Two Properties. Applications to ODE and PDE.
APril 8: Review
Announcements For MATH 305
Back to My Home Page
Homework Four (due: Feb. 7) HW3
Homework Five (due: Feb. 22) HW3
Outline of Teaching Plan For MATH 305
Jan. 10: Fundamentals of complex variable. Euler's formula. Polar coordinate. Principal value of argument Arg (z).
Jan. 12: Arg (z) and arg (z). De Moivre's formula. Roots of unit. Roots of a complex variable.
Jan. 14: Complex exponential. Sets in the complex plane. Functions of complex variables.
Jan. 17: Functions of complex variables. Image under linear and Mobius map $ w=(a+bz)/ (c+dz)$.
Jan. 19: Image under $w=z^2$. Continuous, differentiable, analytic. Cauchy-Riemann equation.
Jan. 21: Consequences of Cauchy-Riemann equation. Harmonic Functions. Conformal Mapping. Level Sets. More notes can be found here. Lecture Notes on Conformal Mappings
Jan. 24: Laplace under analytical mappings. $\partial_{\bar{z}} f(z)=0$. Conformal Mappings. Elementary Functions.
Jan. 26: Elementary functions $ e^z$ and $ \sin (z)$. Images under $ e^z$ and $ sin (z)$.
Jan. 28: Properties of $\sin z$ and $ sinh (z)$. Introduction of $Log (z)$.
Jan. 31: Multi-valued functions. introduction of $log (z)$ and $Log (z)$ and their properties.
Feb. 2: Multi-valued functions. Introduction of $z^\alpha$ and branch cuts.
Feb. 4: Multi-valued functions. Branch cuts for $ (z^2-1)^{\frac{1}{2}}$.
Feb. 7: Branch Cuts for $ (z^3-z)^{1/2}, (z^3-z)^{1/3}, (z^2+1)^{1/2}$.
Feb. 9: Inverse function of sin (z). Solving Laplace equation with Arg (z)
Feb. 11: Midterm 1
Feb. 14: Complex integrals. Contours (Paths).
Feb. 16: Fundamental Theorem of Calculus in the Complex Case. Examples.
Feb. 18: Cauchy-Coursat Theorem. simply-connected domains. Path independence and deformation of path.
Feb. 21-25: Spring Break
Feb. 28: Path Independence. Cauchy Integral Formula. Examples.
Mar. 2: Applications of Cauchy Integral Formula. Computation of real integrals.
Mar. 4: Consequences of Cauchy Integral Formula. Functions with finite order singularity.
Mar. 7: Consequences of Cauchy Integral Formula. Liouville Theorem: bounded entire functions are constants.
March 9: Maximum Modulus Principle.
March 11: Argument Principle,Rouche's Theorem (Lecture Note 6-5).
March 14: Nyquist criterion, applications to ODE.
March 16: Nyquist criterion. Introduction to Laurant series .(Lecture Note 7).
March 18:Introduction to Laurant series .(Lecture Note 7). Classification of singularities and computations of residues. See explanations of the notation O(...) here Notation $O(z^m)$
March 21: Lecture Note 7. Cauchy Residue Theorem. Computation of residues and contour integrals.
March 23: Applications of Cauchy Residue Theorem. Lecture Note 8, page 1-3.
March 25: Midterm 2
March 28: Type I, Type II real integrals.
March 30: Type III real integrals.
April 1: Type IV and Type V integrals. Integrals involving Multi-valued functions.
April 4: Type V integrals. Finish lecture 8.
April 6 (last class): Fourier transforms and inverse Fourier transforms. Two Properties. Applications to ODE and PDE.
APril 8: Review
Announcements For MATH 305
Back to My Home Page
Homework Five (due: Feb. 22) HW3
Outline of Teaching Plan For MATH 305
Jan. 10: Fundamentals of complex variable. Euler's formula. Polar coordinate. Principal value of argument Arg (z).
Jan. 12: Arg (z) and arg (z). De Moivre's formula. Roots of unit. Roots of a complex variable.
Jan. 14: Complex exponential. Sets in the complex plane. Functions of complex variables.
Jan. 17: Functions of complex variables. Image under linear and Mobius map $ w=(a+bz)/ (c+dz)$.
Jan. 19: Image under $w=z^2$. Continuous, differentiable, analytic. Cauchy-Riemann equation.
Jan. 21: Consequences of Cauchy-Riemann equation. Harmonic Functions. Conformal Mapping. Level Sets. More notes can be found here. Lecture Notes on Conformal Mappings
Jan. 24: Laplace under analytical mappings. $\partial_{\bar{z}} f(z)=0$. Conformal Mappings. Elementary Functions.
Jan. 26: Elementary functions $ e^z$ and $ \sin (z)$. Images under $ e^z$ and $ sin (z)$.
Jan. 28: Properties of $\sin z$ and $ sinh (z)$. Introduction of $Log (z)$.
Jan. 31: Multi-valued functions. introduction of $log (z)$ and $Log (z)$ and their properties.
Feb. 2: Multi-valued functions. Introduction of $z^\alpha$ and branch cuts.
Feb. 4: Multi-valued functions. Branch cuts for $ (z^2-1)^{\frac{1}{2}}$.
Feb. 7: Branch Cuts for $ (z^3-z)^{1/2}, (z^3-z)^{1/3}, (z^2+1)^{1/2}$.
Feb. 9: Inverse function of sin (z). Solving Laplace equation with Arg (z)
Feb. 11: Midterm 1
Feb. 14: Complex integrals. Contours (Paths).
Feb. 16: Fundamental Theorem of Calculus in the Complex Case. Examples.
Feb. 18: Cauchy-Coursat Theorem. simply-connected domains. Path independence and deformation of path.
Feb. 21-25: Spring Break
Feb. 28: Path Independence. Cauchy Integral Formula. Examples.
Mar. 2: Applications of Cauchy Integral Formula. Computation of real integrals.
Mar. 4: Consequences of Cauchy Integral Formula. Functions with finite order singularity.
Mar. 7: Consequences of Cauchy Integral Formula. Liouville Theorem: bounded entire functions are constants.
March 9: Maximum Modulus Principle.
March 11: Argument Principle,Rouche's Theorem (Lecture Note 6-5).
March 14: Nyquist criterion, applications to ODE.
March 16: Nyquist criterion. Introduction to Laurant series .(Lecture Note 7).
March 18:Introduction to Laurant series .(Lecture Note 7). Classification of singularities and computations of residues. See explanations of the notation O(...) here Notation $O(z^m)$
March 21: Lecture Note 7. Cauchy Residue Theorem. Computation of residues and contour integrals.
March 23: Applications of Cauchy Residue Theorem. Lecture Note 8, page 1-3.
March 25: Midterm 2
March 28: Type I, Type II real integrals.
March 30: Type III real integrals.
April 1: Type IV and Type V integrals. Integrals involving Multi-valued functions.
April 4: Type V integrals. Finish lecture 8.
April 6 (last class): Fourier transforms and inverse Fourier transforms. Two Properties. Applications to ODE and PDE.
APril 8: Review
Announcements For MATH 305
Back to My Home Page
Outline of Teaching Plan For MATH 305
Jan. 10: Fundamentals of complex variable. Euler's formula. Polar coordinate. Principal value of argument Arg (z).
Jan. 12: Arg (z) and arg (z). De Moivre's formula. Roots of unit. Roots of a complex variable.
Jan. 14: Complex exponential. Sets in the complex plane. Functions of complex variables.
Jan. 17: Functions of complex variables. Image under linear and Mobius map $ w=(a+bz)/ (c+dz)$.
Jan. 19: Image under $w=z^2$. Continuous, differentiable, analytic. Cauchy-Riemann equation.
Jan. 21: Consequences of Cauchy-Riemann equation. Harmonic Functions. Conformal Mapping. Level Sets. More notes can be found here. Lecture Notes on Conformal Mappings
Jan. 24: Laplace under analytical mappings. $\partial_{\bar{z}} f(z)=0$. Conformal Mappings. Elementary Functions.
Jan. 26: Elementary functions $ e^z$ and $ \sin (z)$. Images under $ e^z$ and $ sin (z)$.
Jan. 28: Properties of $\sin z$ and $ sinh (z)$. Introduction of $Log (z)$.
Jan. 31: Multi-valued functions. introduction of $log (z)$ and $Log (z)$ and their properties.
Feb. 2: Multi-valued functions. Introduction of $z^\alpha$ and branch cuts.
Feb. 4: Multi-valued functions. Branch cuts for $ (z^2-1)^{\frac{1}{2}}$.
Feb. 7: Branch Cuts for $ (z^3-z)^{1/2}, (z^3-z)^{1/3}, (z^2+1)^{1/2}$.
Feb. 9: Inverse function of sin (z). Solving Laplace equation with Arg (z)
Feb. 11: Midterm 1
Feb. 14: Complex integrals. Contours (Paths).
Feb. 16: Fundamental Theorem of Calculus in the Complex Case. Examples.
Feb. 18: Cauchy-Coursat Theorem. simply-connected domains. Path independence and deformation of path.
Feb. 21-25: Spring Break
Feb. 28: Path Independence. Cauchy Integral Formula. Examples.
Mar. 2: Applications of Cauchy Integral Formula. Computation of real integrals.
Mar. 4: Consequences of Cauchy Integral Formula. Functions with finite order singularity.
Mar. 7: Consequences of Cauchy Integral Formula. Liouville Theorem: bounded entire functions are constants.
March 9: Maximum Modulus Principle.
March 11: Argument Principle,Rouche's Theorem (Lecture Note 6-5).
March 14: Nyquist criterion, applications to ODE.
March 16: Nyquist criterion. Introduction to Laurant series .(Lecture Note 7).
March 18:Introduction to Laurant series .(Lecture Note 7). Classification of singularities and computations of residues. See explanations of the notation O(...) here Notation $O(z^m)$
March 21: Lecture Note 7. Cauchy Residue Theorem. Computation of residues and contour integrals.
March 23: Applications of Cauchy Residue Theorem. Lecture Note 8, page 1-3.
March 25: Midterm 2
March 28: Type I, Type II real integrals.
March 30: Type III real integrals.
April 1: Type IV and Type V integrals. Integrals involving Multi-valued functions.
April 4: Type V integrals. Finish lecture 8.
April 6 (last class): Fourier transforms and inverse Fourier transforms. Two Properties. Applications to ODE and PDE.
APril 8: Review
Announcements For MATH 305
Back to My Home Page
Jan. 12: Arg (z) and arg (z). De Moivre's formula. Roots of unit. Roots of a complex variable.
Jan. 14: Complex exponential. Sets in the complex plane. Functions of complex variables.
Jan. 17: Functions of complex variables. Image under linear and Mobius map $ w=(a+bz)/ (c+dz)$.
Jan. 19: Image under $w=z^2$. Continuous, differentiable, analytic. Cauchy-Riemann equation.
Jan. 21: Consequences of Cauchy-Riemann equation. Harmonic Functions. Conformal Mapping. Level Sets. More notes can be found here. Lecture Notes on Conformal Mappings
Jan. 24: Laplace under analytical mappings. $\partial_{\bar{z}} f(z)=0$. Conformal Mappings. Elementary Functions.
Jan. 26: Elementary functions $ e^z$ and $ \sin (z)$. Images under $ e^z$ and $ sin (z)$.
Jan. 28: Properties of $\sin z$ and $ sinh (z)$. Introduction of $Log (z)$.
Jan. 31: Multi-valued functions. introduction of $log (z)$ and $Log (z)$ and their properties.
Feb. 2: Multi-valued functions. Introduction of $z^\alpha$ and branch cuts.
Feb. 4: Multi-valued functions. Branch cuts for $ (z^2-1)^{\frac{1}{2}}$.
Feb. 7: Branch Cuts for $ (z^3-z)^{1/2}, (z^3-z)^{1/3}, (z^2+1)^{1/2}$.
Feb. 9: Inverse function of sin (z). Solving Laplace equation with Arg (z)
Feb. 11: Midterm 1
Feb. 14: Complex integrals. Contours (Paths).
Feb. 16: Fundamental Theorem of Calculus in the Complex Case. Examples.
Feb. 18: Cauchy-Coursat Theorem. simply-connected domains. Path independence and deformation of path.
Feb. 21-25: Spring Break
Feb. 28: Path Independence. Cauchy Integral Formula. Examples.
Mar. 2: Applications of Cauchy Integral Formula. Computation of real integrals.
Mar. 4: Consequences of Cauchy Integral Formula. Functions with finite order singularity.
Mar. 7: Consequences of Cauchy Integral Formula. Liouville Theorem: bounded entire functions are constants.
March 9: Maximum Modulus Principle.
March 11: Argument Principle,Rouche's Theorem (Lecture Note 6-5).
March 14: Nyquist criterion, applications to ODE.
March 16: Nyquist criterion. Introduction to Laurant series .(Lecture Note 7).
March 18:Introduction to Laurant series .(Lecture Note 7). Classification of singularities and computations of residues. See explanations of the notation O(...) here Notation $O(z^m)$
March 21: Lecture Note 7. Cauchy Residue Theorem. Computation of residues and contour integrals.
March 23: Applications of Cauchy Residue Theorem. Lecture Note 8, page 1-3.
March 25: Midterm 2
March 28: Type I, Type II real integrals.
March 30: Type III real integrals.
April 1: Type IV and Type V integrals. Integrals involving Multi-valued functions.
April 4: Type V integrals. Finish lecture 8.
April 6 (last class): Fourier transforms and inverse Fourier transforms. Two Properties. Applications to ODE and PDE.
APril 8: Review
Announcements For MATH 305
Back to My Home Page
Jan. 14: Complex exponential. Sets in the complex plane. Functions of complex variables.
Jan. 17: Functions of complex variables. Image under linear and Mobius map $ w=(a+bz)/ (c+dz)$.
Jan. 19: Image under $w=z^2$. Continuous, differentiable, analytic. Cauchy-Riemann equation.
Jan. 21: Consequences of Cauchy-Riemann equation. Harmonic Functions. Conformal Mapping. Level Sets. More notes can be found here. Lecture Notes on Conformal Mappings
Jan. 24: Laplace under analytical mappings. $\partial_{\bar{z}} f(z)=0$. Conformal Mappings. Elementary Functions.
Jan. 26: Elementary functions $ e^z$ and $ \sin (z)$. Images under $ e^z$ and $ sin (z)$.
Jan. 28: Properties of $\sin z$ and $ sinh (z)$. Introduction of $Log (z)$.
Jan. 31: Multi-valued functions. introduction of $log (z)$ and $Log (z)$ and their properties.
Feb. 2: Multi-valued functions. Introduction of $z^\alpha$ and branch cuts.
Feb. 4: Multi-valued functions. Branch cuts for $ (z^2-1)^{\frac{1}{2}}$.
Feb. 7: Branch Cuts for $ (z^3-z)^{1/2}, (z^3-z)^{1/3}, (z^2+1)^{1/2}$.
Feb. 9: Inverse function of sin (z). Solving Laplace equation with Arg (z)
Feb. 11: Midterm 1
Feb. 14: Complex integrals. Contours (Paths).
Feb. 16: Fundamental Theorem of Calculus in the Complex Case. Examples.
Feb. 18: Cauchy-Coursat Theorem. simply-connected domains. Path independence and deformation of path.
Feb. 21-25: Spring Break
Feb. 28: Path Independence. Cauchy Integral Formula. Examples.
Mar. 2: Applications of Cauchy Integral Formula. Computation of real integrals.
Mar. 4: Consequences of Cauchy Integral Formula. Functions with finite order singularity.
Mar. 7: Consequences of Cauchy Integral Formula. Liouville Theorem: bounded entire functions are constants.
March 9: Maximum Modulus Principle.
March 11: Argument Principle,Rouche's Theorem (Lecture Note 6-5).
March 14: Nyquist criterion, applications to ODE.
March 16: Nyquist criterion. Introduction to Laurant series .(Lecture Note 7).
March 18:Introduction to Laurant series .(Lecture Note 7). Classification of singularities and computations of residues. See explanations of the notation O(...) here Notation $O(z^m)$
March 21: Lecture Note 7. Cauchy Residue Theorem. Computation of residues and contour integrals.
March 23: Applications of Cauchy Residue Theorem. Lecture Note 8, page 1-3.
March 25: Midterm 2
March 28: Type I, Type II real integrals.
March 30: Type III real integrals.
April 1: Type IV and Type V integrals. Integrals involving Multi-valued functions.
April 4: Type V integrals. Finish lecture 8.
April 6 (last class): Fourier transforms and inverse Fourier transforms. Two Properties. Applications to ODE and PDE.
APril 8: Review
Announcements For MATH 305
Back to My Home Page
Jan. 17: Functions of complex variables. Image under linear and Mobius map $ w=(a+bz)/ (c+dz)$.
Jan. 19: Image under $w=z^2$. Continuous, differentiable, analytic. Cauchy-Riemann equation.
Jan. 21: Consequences of Cauchy-Riemann equation. Harmonic Functions. Conformal Mapping. Level Sets. More notes can be found here. Lecture Notes on Conformal Mappings
Jan. 24: Laplace under analytical mappings. $\partial_{\bar{z}} f(z)=0$. Conformal Mappings. Elementary Functions.
Jan. 26: Elementary functions $ e^z$ and $ \sin (z)$. Images under $ e^z$ and $ sin (z)$.
Jan. 28: Properties of $\sin z$ and $ sinh (z)$. Introduction of $Log (z)$.
Jan. 31: Multi-valued functions. introduction of $log (z)$ and $Log (z)$ and their properties.
Feb. 2: Multi-valued functions. Introduction of $z^\alpha$ and branch cuts.
Feb. 4: Multi-valued functions. Branch cuts for $ (z^2-1)^{\frac{1}{2}}$.
Feb. 7: Branch Cuts for $ (z^3-z)^{1/2}, (z^3-z)^{1/3}, (z^2+1)^{1/2}$.
Feb. 9: Inverse function of sin (z). Solving Laplace equation with Arg (z)
Feb. 11: Midterm 1
Feb. 14: Complex integrals. Contours (Paths).
Feb. 16: Fundamental Theorem of Calculus in the Complex Case. Examples.
Feb. 18: Cauchy-Coursat Theorem. simply-connected domains. Path independence and deformation of path.
Feb. 21-25: Spring Break
Feb. 28: Path Independence. Cauchy Integral Formula. Examples.
Mar. 2: Applications of Cauchy Integral Formula. Computation of real integrals.
Mar. 4: Consequences of Cauchy Integral Formula. Functions with finite order singularity.
Mar. 7: Consequences of Cauchy Integral Formula. Liouville Theorem: bounded entire functions are constants.
March 9: Maximum Modulus Principle.
March 11: Argument Principle,Rouche's Theorem (Lecture Note 6-5).
March 14: Nyquist criterion, applications to ODE.
March 16: Nyquist criterion. Introduction to Laurant series .(Lecture Note 7).
March 18:Introduction to Laurant series .(Lecture Note 7). Classification of singularities and computations of residues. See explanations of the notation O(...) here Notation $O(z^m)$
March 21: Lecture Note 7. Cauchy Residue Theorem. Computation of residues and contour integrals.
March 23: Applications of Cauchy Residue Theorem. Lecture Note 8, page 1-3.
March 25: Midterm 2
March 28: Type I, Type II real integrals.
March 30: Type III real integrals.
April 1: Type IV and Type V integrals. Integrals involving Multi-valued functions.
April 4: Type V integrals. Finish lecture 8.
April 6 (last class): Fourier transforms and inverse Fourier transforms. Two Properties. Applications to ODE and PDE.
APril 8: Review
Announcements For MATH 305
Back to My Home Page
Jan. 19: Image under $w=z^2$. Continuous, differentiable, analytic. Cauchy-Riemann equation.
Jan. 21: Consequences of Cauchy-Riemann equation. Harmonic Functions. Conformal Mapping. Level Sets. More notes can be found here. Lecture Notes on Conformal Mappings
Jan. 24: Laplace under analytical mappings. $\partial_{\bar{z}} f(z)=0$. Conformal Mappings. Elementary Functions.
Jan. 26: Elementary functions $ e^z$ and $ \sin (z)$. Images under $ e^z$ and $ sin (z)$.
Jan. 28: Properties of $\sin z$ and $ sinh (z)$. Introduction of $Log (z)$.
Jan. 31: Multi-valued functions. introduction of $log (z)$ and $Log (z)$ and their properties.
Feb. 2: Multi-valued functions. Introduction of $z^\alpha$ and branch cuts.
Feb. 4: Multi-valued functions. Branch cuts for $ (z^2-1)^{\frac{1}{2}}$.
Feb. 7: Branch Cuts for $ (z^3-z)^{1/2}, (z^3-z)^{1/3}, (z^2+1)^{1/2}$.
Feb. 9: Inverse function of sin (z). Solving Laplace equation with Arg (z)
Feb. 11: Midterm 1
Feb. 14: Complex integrals. Contours (Paths).
Feb. 16: Fundamental Theorem of Calculus in the Complex Case. Examples.
Feb. 18: Cauchy-Coursat Theorem. simply-connected domains. Path independence and deformation of path.
Feb. 21-25: Spring Break
Feb. 28: Path Independence. Cauchy Integral Formula. Examples.
Mar. 2: Applications of Cauchy Integral Formula. Computation of real integrals.
Mar. 4: Consequences of Cauchy Integral Formula. Functions with finite order singularity.
Mar. 7: Consequences of Cauchy Integral Formula. Liouville Theorem: bounded entire functions are constants.
March 9: Maximum Modulus Principle.
March 11: Argument Principle,Rouche's Theorem (Lecture Note 6-5).
March 14: Nyquist criterion, applications to ODE.
March 16: Nyquist criterion. Introduction to Laurant series .(Lecture Note 7).
March 18:Introduction to Laurant series .(Lecture Note 7). Classification of singularities and computations of residues. See explanations of the notation O(...) here Notation $O(z^m)$
March 21: Lecture Note 7. Cauchy Residue Theorem. Computation of residues and contour integrals.
March 23: Applications of Cauchy Residue Theorem. Lecture Note 8, page 1-3.
March 25: Midterm 2
March 28: Type I, Type II real integrals.
March 30: Type III real integrals.
April 1: Type IV and Type V integrals. Integrals involving Multi-valued functions.
April 4: Type V integrals. Finish lecture 8.
April 6 (last class): Fourier transforms and inverse Fourier transforms. Two Properties. Applications to ODE and PDE.
APril 8: Review
Announcements For MATH 305
Back to My Home Page
Jan. 21: Consequences of Cauchy-Riemann equation. Harmonic Functions. Conformal Mapping. Level Sets. More notes can be found here. Lecture Notes on Conformal Mappings
Jan. 24: Laplace under analytical mappings. $\partial_{\bar{z}} f(z)=0$. Conformal Mappings. Elementary Functions.
Jan. 26: Elementary functions $ e^z$ and $ \sin (z)$. Images under $ e^z$ and $ sin (z)$.
Jan. 28: Properties of $\sin z$ and $ sinh (z)$. Introduction of $Log (z)$.
Jan. 31: Multi-valued functions. introduction of $log (z)$ and $Log (z)$ and their properties.
Feb. 2: Multi-valued functions. Introduction of $z^\alpha$ and branch cuts.
Feb. 4: Multi-valued functions. Branch cuts for $ (z^2-1)^{\frac{1}{2}}$.
Feb. 7: Branch Cuts for $ (z^3-z)^{1/2}, (z^3-z)^{1/3}, (z^2+1)^{1/2}$.
Feb. 9: Inverse function of sin (z). Solving Laplace equation with Arg (z)
Feb. 11: Midterm 1
Feb. 14: Complex integrals. Contours (Paths).
Feb. 16: Fundamental Theorem of Calculus in the Complex Case. Examples.
Feb. 18: Cauchy-Coursat Theorem. simply-connected domains. Path independence and deformation of path.
Feb. 21-25: Spring Break
Feb. 28: Path Independence. Cauchy Integral Formula. Examples.
Mar. 2: Applications of Cauchy Integral Formula. Computation of real integrals.
Mar. 4: Consequences of Cauchy Integral Formula. Functions with finite order singularity.
Mar. 7: Consequences of Cauchy Integral Formula. Liouville Theorem: bounded entire functions are constants.
March 9: Maximum Modulus Principle.
March 11: Argument Principle,Rouche's Theorem (Lecture Note 6-5).
March 14: Nyquist criterion, applications to ODE.
March 16: Nyquist criterion. Introduction to Laurant series .(Lecture Note 7).
March 18:Introduction to Laurant series .(Lecture Note 7). Classification of singularities and computations of residues. See explanations of the notation O(...) here Notation $O(z^m)$
March 21: Lecture Note 7. Cauchy Residue Theorem. Computation of residues and contour integrals.
March 23: Applications of Cauchy Residue Theorem. Lecture Note 8, page 1-3.
March 25: Midterm 2
March 28: Type I, Type II real integrals.
March 30: Type III real integrals.
April 1: Type IV and Type V integrals. Integrals involving Multi-valued functions.
April 4: Type V integrals. Finish lecture 8.
April 6 (last class): Fourier transforms and inverse Fourier transforms. Two Properties. Applications to ODE and PDE.
APril 8: Review
Announcements For MATH 305
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Jan. 24: Laplace under analytical mappings. $\partial_{\bar{z}} f(z)=0$. Conformal Mappings. Elementary Functions.
Jan. 26: Elementary functions $ e^z$ and $ \sin (z)$. Images under $ e^z$ and $ sin (z)$.
Jan. 28: Properties of $\sin z$ and $ sinh (z)$. Introduction of $Log (z)$.
Jan. 31: Multi-valued functions. introduction of $log (z)$ and $Log (z)$ and their properties.
Feb. 2: Multi-valued functions. Introduction of $z^\alpha$ and branch cuts.
Feb. 4: Multi-valued functions. Branch cuts for $ (z^2-1)^{\frac{1}{2}}$.
Feb. 7: Branch Cuts for $ (z^3-z)^{1/2}, (z^3-z)^{1/3}, (z^2+1)^{1/2}$.
Feb. 9: Inverse function of sin (z). Solving Laplace equation with Arg (z)
Feb. 11: Midterm 1
Feb. 14: Complex integrals. Contours (Paths).
Feb. 16: Fundamental Theorem of Calculus in the Complex Case. Examples.
Feb. 18: Cauchy-Coursat Theorem. simply-connected domains. Path independence and deformation of path.
Feb. 21-25: Spring Break
Feb. 28: Path Independence. Cauchy Integral Formula. Examples.
Mar. 2: Applications of Cauchy Integral Formula. Computation of real integrals.
Mar. 4: Consequences of Cauchy Integral Formula. Functions with finite order singularity.
Mar. 7: Consequences of Cauchy Integral Formula. Liouville Theorem: bounded entire functions are constants.
March 9: Maximum Modulus Principle.
March 11: Argument Principle,Rouche's Theorem (Lecture Note 6-5).
March 14: Nyquist criterion, applications to ODE.
March 16: Nyquist criterion. Introduction to Laurant series .(Lecture Note 7).
March 18:Introduction to Laurant series .(Lecture Note 7). Classification of singularities and computations of residues. See explanations of the notation O(...) here Notation $O(z^m)$
March 21: Lecture Note 7. Cauchy Residue Theorem. Computation of residues and contour integrals.
March 23: Applications of Cauchy Residue Theorem. Lecture Note 8, page 1-3.
March 25: Midterm 2
March 28: Type I, Type II real integrals.
March 30: Type III real integrals.
April 1: Type IV and Type V integrals. Integrals involving Multi-valued functions.
April 4: Type V integrals. Finish lecture 8.
April 6 (last class): Fourier transforms and inverse Fourier transforms. Two Properties. Applications to ODE and PDE.
APril 8: Review
Announcements For MATH 305
Back to My Home Page
Jan. 26: Elementary functions $ e^z$ and $ \sin (z)$. Images under $ e^z$ and $ sin (z)$.
Jan. 28: Properties of $\sin z$ and $ sinh (z)$. Introduction of $Log (z)$.
Jan. 31: Multi-valued functions. introduction of $log (z)$ and $Log (z)$ and their properties.
Feb. 2: Multi-valued functions. Introduction of $z^\alpha$ and branch cuts.
Feb. 4: Multi-valued functions. Branch cuts for $ (z^2-1)^{\frac{1}{2}}$.
Feb. 7: Branch Cuts for $ (z^3-z)^{1/2}, (z^3-z)^{1/3}, (z^2+1)^{1/2}$.
Feb. 9: Inverse function of sin (z). Solving Laplace equation with Arg (z)
Feb. 11: Midterm 1
Feb. 14: Complex integrals. Contours (Paths).
Feb. 16: Fundamental Theorem of Calculus in the Complex Case. Examples.
Feb. 18: Cauchy-Coursat Theorem. simply-connected domains. Path independence and deformation of path.
Feb. 21-25: Spring Break
Feb. 28: Path Independence. Cauchy Integral Formula. Examples.
Mar. 2: Applications of Cauchy Integral Formula. Computation of real integrals.
Mar. 4: Consequences of Cauchy Integral Formula. Functions with finite order singularity.
Mar. 7: Consequences of Cauchy Integral Formula. Liouville Theorem: bounded entire functions are constants.
March 9: Maximum Modulus Principle.
March 11: Argument Principle,Rouche's Theorem (Lecture Note 6-5).
March 14: Nyquist criterion, applications to ODE.
March 16: Nyquist criterion. Introduction to Laurant series .(Lecture Note 7).
March 18:Introduction to Laurant series .(Lecture Note 7). Classification of singularities and computations of residues. See explanations of the notation O(...) here Notation $O(z^m)$
March 21: Lecture Note 7. Cauchy Residue Theorem. Computation of residues and contour integrals.
March 23: Applications of Cauchy Residue Theorem. Lecture Note 8, page 1-3.
March 25: Midterm 2
March 28: Type I, Type II real integrals.
March 30: Type III real integrals.
April 1: Type IV and Type V integrals. Integrals involving Multi-valued functions.
April 4: Type V integrals. Finish lecture 8.
April 6 (last class): Fourier transforms and inverse Fourier transforms. Two Properties. Applications to ODE and PDE.
APril 8: Review
Announcements For MATH 305
Back to My Home Page
Jan. 28: Properties of $\sin z$ and $ sinh (z)$. Introduction of $Log (z)$.
Jan. 31: Multi-valued functions. introduction of $log (z)$ and $Log (z)$ and their properties.
Feb. 2: Multi-valued functions. Introduction of $z^\alpha$ and branch cuts.
Feb. 4: Multi-valued functions. Branch cuts for $ (z^2-1)^{\frac{1}{2}}$.
Feb. 7: Branch Cuts for $ (z^3-z)^{1/2}, (z^3-z)^{1/3}, (z^2+1)^{1/2}$.
Feb. 9: Inverse function of sin (z). Solving Laplace equation with Arg (z)
Feb. 11: Midterm 1
Feb. 14: Complex integrals. Contours (Paths).
Feb. 16: Fundamental Theorem of Calculus in the Complex Case. Examples.
Feb. 18: Cauchy-Coursat Theorem. simply-connected domains. Path independence and deformation of path.
Feb. 21-25: Spring Break
Feb. 28: Path Independence. Cauchy Integral Formula. Examples.
Mar. 2: Applications of Cauchy Integral Formula. Computation of real integrals.
Mar. 4: Consequences of Cauchy Integral Formula. Functions with finite order singularity.
Mar. 7: Consequences of Cauchy Integral Formula. Liouville Theorem: bounded entire functions are constants.
March 9: Maximum Modulus Principle.
March 11: Argument Principle,Rouche's Theorem (Lecture Note 6-5).
March 14: Nyquist criterion, applications to ODE.
March 16: Nyquist criterion. Introduction to Laurant series .(Lecture Note 7).
March 18:Introduction to Laurant series .(Lecture Note 7). Classification of singularities and computations of residues. See explanations of the notation O(...) here Notation $O(z^m)$
March 21: Lecture Note 7. Cauchy Residue Theorem. Computation of residues and contour integrals.
March 23: Applications of Cauchy Residue Theorem. Lecture Note 8, page 1-3.
March 25: Midterm 2
March 28: Type I, Type II real integrals.
March 30: Type III real integrals.
April 1: Type IV and Type V integrals. Integrals involving Multi-valued functions.
April 4: Type V integrals. Finish lecture 8.
April 6 (last class): Fourier transforms and inverse Fourier transforms. Two Properties. Applications to ODE and PDE.
APril 8: Review
Announcements For MATH 305
Back to My Home Page
Jan. 31: Multi-valued functions. introduction of $log (z)$ and $Log (z)$ and their properties.
Feb. 2: Multi-valued functions. Introduction of $z^\alpha$ and branch cuts.
Feb. 4: Multi-valued functions. Branch cuts for $ (z^2-1)^{\frac{1}{2}}$.
Feb. 7: Branch Cuts for $ (z^3-z)^{1/2}, (z^3-z)^{1/3}, (z^2+1)^{1/2}$.
Feb. 9: Inverse function of sin (z). Solving Laplace equation with Arg (z)
Feb. 11: Midterm 1
Feb. 14: Complex integrals. Contours (Paths).
Feb. 16: Fundamental Theorem of Calculus in the Complex Case. Examples.
Feb. 18: Cauchy-Coursat Theorem. simply-connected domains. Path independence and deformation of path.
Feb. 21-25: Spring Break
Feb. 28: Path Independence. Cauchy Integral Formula. Examples.
Mar. 2: Applications of Cauchy Integral Formula. Computation of real integrals.
Mar. 4: Consequences of Cauchy Integral Formula. Functions with finite order singularity.
Mar. 7: Consequences of Cauchy Integral Formula. Liouville Theorem: bounded entire functions are constants.
March 9: Maximum Modulus Principle.
March 11: Argument Principle,Rouche's Theorem (Lecture Note 6-5).
March 14: Nyquist criterion, applications to ODE.
March 16: Nyquist criterion. Introduction to Laurant series .(Lecture Note 7).
March 18:Introduction to Laurant series .(Lecture Note 7). Classification of singularities and computations of residues. See explanations of the notation O(...) here Notation $O(z^m)$
March 21: Lecture Note 7. Cauchy Residue Theorem. Computation of residues and contour integrals.
March 23: Applications of Cauchy Residue Theorem. Lecture Note 8, page 1-3.
March 25: Midterm 2
March 28: Type I, Type II real integrals.
March 30: Type III real integrals.
April 1: Type IV and Type V integrals. Integrals involving Multi-valued functions.
April 4: Type V integrals. Finish lecture 8.
April 6 (last class): Fourier transforms and inverse Fourier transforms. Two Properties. Applications to ODE and PDE.
APril 8: Review
Announcements For MATH 305
Back to My Home Page
Feb. 2: Multi-valued functions. Introduction of $z^\alpha$ and branch cuts.
Feb. 4: Multi-valued functions. Branch cuts for $ (z^2-1)^{\frac{1}{2}}$.
Feb. 7: Branch Cuts for $ (z^3-z)^{1/2}, (z^3-z)^{1/3}, (z^2+1)^{1/2}$.
Feb. 9: Inverse function of sin (z). Solving Laplace equation with Arg (z)
Feb. 11: Midterm 1
Feb. 14: Complex integrals. Contours (Paths).
Feb. 16: Fundamental Theorem of Calculus in the Complex Case. Examples.
Feb. 18: Cauchy-Coursat Theorem. simply-connected domains. Path independence and deformation of path.
Feb. 21-25: Spring Break
Feb. 28: Path Independence. Cauchy Integral Formula. Examples.
Mar. 2: Applications of Cauchy Integral Formula. Computation of real integrals.
Mar. 4: Consequences of Cauchy Integral Formula. Functions with finite order singularity.
Mar. 7: Consequences of Cauchy Integral Formula. Liouville Theorem: bounded entire functions are constants.
March 9: Maximum Modulus Principle.
March 11: Argument Principle,Rouche's Theorem (Lecture Note 6-5).
March 14: Nyquist criterion, applications to ODE.
March 16: Nyquist criterion. Introduction to Laurant series .(Lecture Note 7).
March 18:Introduction to Laurant series .(Lecture Note 7). Classification of singularities and computations of residues. See explanations of the notation O(...) here Notation $O(z^m)$
March 21: Lecture Note 7. Cauchy Residue Theorem. Computation of residues and contour integrals.
March 23: Applications of Cauchy Residue Theorem. Lecture Note 8, page 1-3.
March 25: Midterm 2
March 28: Type I, Type II real integrals.
March 30: Type III real integrals.
April 1: Type IV and Type V integrals. Integrals involving Multi-valued functions.
April 4: Type V integrals. Finish lecture 8.
April 6 (last class): Fourier transforms and inverse Fourier transforms. Two Properties. Applications to ODE and PDE.
APril 8: Review
Announcements For MATH 305
Back to My Home Page
Feb. 4: Multi-valued functions. Branch cuts for $ (z^2-1)^{\frac{1}{2}}$.
Feb. 7: Branch Cuts for $ (z^3-z)^{1/2}, (z^3-z)^{1/3}, (z^2+1)^{1/2}$.
Feb. 9: Inverse function of sin (z). Solving Laplace equation with Arg (z)
Feb. 11: Midterm 1
Feb. 14: Complex integrals. Contours (Paths).
Feb. 16: Fundamental Theorem of Calculus in the Complex Case. Examples.
Feb. 18: Cauchy-Coursat Theorem. simply-connected domains. Path independence and deformation of path.
Feb. 21-25: Spring Break
Feb. 28: Path Independence. Cauchy Integral Formula. Examples.
Mar. 2: Applications of Cauchy Integral Formula. Computation of real integrals.
Mar. 4: Consequences of Cauchy Integral Formula. Functions with finite order singularity.
Mar. 7: Consequences of Cauchy Integral Formula. Liouville Theorem: bounded entire functions are constants.
March 9: Maximum Modulus Principle.
March 11: Argument Principle,Rouche's Theorem (Lecture Note 6-5).
March 14: Nyquist criterion, applications to ODE.
March 16: Nyquist criterion. Introduction to Laurant series .(Lecture Note 7).
March 18:Introduction to Laurant series .(Lecture Note 7). Classification of singularities and computations of residues. See explanations of the notation O(...) here Notation $O(z^m)$
March 21: Lecture Note 7. Cauchy Residue Theorem. Computation of residues and contour integrals.
March 23: Applications of Cauchy Residue Theorem. Lecture Note 8, page 1-3.
March 25: Midterm 2
March 28: Type I, Type II real integrals.
March 30: Type III real integrals.
April 1: Type IV and Type V integrals. Integrals involving Multi-valued functions.
April 4: Type V integrals. Finish lecture 8.
April 6 (last class): Fourier transforms and inverse Fourier transforms. Two Properties. Applications to ODE and PDE.
APril 8: Review
Announcements For MATH 305
Back to My Home Page
Feb. 7: Branch Cuts for $ (z^3-z)^{1/2}, (z^3-z)^{1/3}, (z^2+1)^{1/2}$.
Feb. 9: Inverse function of sin (z). Solving Laplace equation with Arg (z)
Feb. 11: Midterm 1
Feb. 14: Complex integrals. Contours (Paths).
Feb. 16: Fundamental Theorem of Calculus in the Complex Case. Examples.
Feb. 18: Cauchy-Coursat Theorem. simply-connected domains. Path independence and deformation of path.
Feb. 21-25: Spring Break
Feb. 28: Path Independence. Cauchy Integral Formula. Examples.
Mar. 2: Applications of Cauchy Integral Formula. Computation of real integrals.
Mar. 4: Consequences of Cauchy Integral Formula. Functions with finite order singularity.
Mar. 7: Consequences of Cauchy Integral Formula. Liouville Theorem: bounded entire functions are constants.
March 9: Maximum Modulus Principle.
March 11: Argument Principle,Rouche's Theorem (Lecture Note 6-5).
March 14: Nyquist criterion, applications to ODE.
March 16: Nyquist criterion. Introduction to Laurant series .(Lecture Note 7).
March 18:Introduction to Laurant series .(Lecture Note 7). Classification of singularities and computations of residues. See explanations of the notation O(...) here Notation $O(z^m)$
March 21: Lecture Note 7. Cauchy Residue Theorem. Computation of residues and contour integrals.
March 23: Applications of Cauchy Residue Theorem. Lecture Note 8, page 1-3.
March 25: Midterm 2
March 28: Type I, Type II real integrals.
March 30: Type III real integrals.
April 1: Type IV and Type V integrals. Integrals involving Multi-valued functions.
April 4: Type V integrals. Finish lecture 8.
April 6 (last class): Fourier transforms and inverse Fourier transforms. Two Properties. Applications to ODE and PDE.
APril 8: Review
Announcements For MATH 305
Back to My Home Page
Feb. 9: Inverse function of sin (z). Solving Laplace equation with Arg (z)
Feb. 11: Midterm 1
Feb. 14: Complex integrals. Contours (Paths).
Feb. 16: Fundamental Theorem of Calculus in the Complex Case. Examples.
Feb. 18: Cauchy-Coursat Theorem. simply-connected domains. Path independence and deformation of path.
Feb. 21-25: Spring Break
Feb. 28: Path Independence. Cauchy Integral Formula. Examples.
Mar. 2: Applications of Cauchy Integral Formula. Computation of real integrals.
Mar. 4: Consequences of Cauchy Integral Formula. Functions with finite order singularity.
Mar. 7: Consequences of Cauchy Integral Formula. Liouville Theorem: bounded entire functions are constants.
March 9: Maximum Modulus Principle.
March 11: Argument Principle,Rouche's Theorem (Lecture Note 6-5).
March 14: Nyquist criterion, applications to ODE.
March 16: Nyquist criterion. Introduction to Laurant series .(Lecture Note 7).
March 18:Introduction to Laurant series .(Lecture Note 7). Classification of singularities and computations of residues. See explanations of the notation O(...) here Notation $O(z^m)$
March 21: Lecture Note 7. Cauchy Residue Theorem. Computation of residues and contour integrals.
March 23: Applications of Cauchy Residue Theorem. Lecture Note 8, page 1-3.
March 25: Midterm 2
March 28: Type I, Type II real integrals.
March 30: Type III real integrals.
April 1: Type IV and Type V integrals. Integrals involving Multi-valued functions.
April 4: Type V integrals. Finish lecture 8.
April 6 (last class): Fourier transforms and inverse Fourier transforms. Two Properties. Applications to ODE and PDE.
APril 8: Review
Announcements For MATH 305
Back to My Home Page
Feb. 11: Midterm 1
Feb. 14: Complex integrals. Contours (Paths).
Feb. 16: Fundamental Theorem of Calculus in the Complex Case. Examples.
Feb. 18: Cauchy-Coursat Theorem. simply-connected domains. Path independence and deformation of path.
Feb. 21-25: Spring Break
Feb. 28: Path Independence. Cauchy Integral Formula. Examples.
Mar. 2: Applications of Cauchy Integral Formula. Computation of real integrals.
Mar. 4: Consequences of Cauchy Integral Formula. Functions with finite order singularity.
Mar. 7: Consequences of Cauchy Integral Formula. Liouville Theorem: bounded entire functions are constants.
March 9: Maximum Modulus Principle.
March 11: Argument Principle,Rouche's Theorem (Lecture Note 6-5).
March 14: Nyquist criterion, applications to ODE.
March 16: Nyquist criterion. Introduction to Laurant series .(Lecture Note 7).
March 18:Introduction to Laurant series .(Lecture Note 7). Classification of singularities and computations of residues. See explanations of the notation O(...) here Notation $O(z^m)$
March 21: Lecture Note 7. Cauchy Residue Theorem. Computation of residues and contour integrals.
March 23: Applications of Cauchy Residue Theorem. Lecture Note 8, page 1-3.
March 25: Midterm 2
March 28: Type I, Type II real integrals.
March 30: Type III real integrals.
April 1: Type IV and Type V integrals. Integrals involving Multi-valued functions.
April 4: Type V integrals. Finish lecture 8.
April 6 (last class): Fourier transforms and inverse Fourier transforms. Two Properties. Applications to ODE and PDE.
APril 8: Review
Announcements For MATH 305
Back to My Home Page
Feb. 14: Complex integrals. Contours (Paths).
Feb. 16: Fundamental Theorem of Calculus in the Complex Case. Examples.
Feb. 18: Cauchy-Coursat Theorem. simply-connected domains. Path independence and deformation of path.
Feb. 21-25: Spring Break
Feb. 28: Path Independence. Cauchy Integral Formula. Examples.
Mar. 2: Applications of Cauchy Integral Formula. Computation of real integrals.
Mar. 4: Consequences of Cauchy Integral Formula. Functions with finite order singularity.
Mar. 7: Consequences of Cauchy Integral Formula. Liouville Theorem: bounded entire functions are constants.
March 9: Maximum Modulus Principle.
March 11: Argument Principle,Rouche's Theorem (Lecture Note 6-5).
March 14: Nyquist criterion, applications to ODE.
March 16: Nyquist criterion. Introduction to Laurant series .(Lecture Note 7).
March 18:Introduction to Laurant series .(Lecture Note 7). Classification of singularities and computations of residues. See explanations of the notation O(...) here Notation $O(z^m)$
March 21: Lecture Note 7. Cauchy Residue Theorem. Computation of residues and contour integrals.
March 23: Applications of Cauchy Residue Theorem. Lecture Note 8, page 1-3.
March 25: Midterm 2
March 28: Type I, Type II real integrals.
March 30: Type III real integrals.
April 1: Type IV and Type V integrals. Integrals involving Multi-valued functions.
April 4: Type V integrals. Finish lecture 8.
April 6 (last class): Fourier transforms and inverse Fourier transforms. Two Properties. Applications to ODE and PDE.
APril 8: Review
Announcements For MATH 305
Back to My Home Page
Feb. 16: Fundamental Theorem of Calculus in the Complex Case. Examples.
Feb. 18: Cauchy-Coursat Theorem. simply-connected domains. Path independence and deformation of path.
Feb. 21-25: Spring Break
Feb. 28: Path Independence. Cauchy Integral Formula. Examples.
Mar. 2: Applications of Cauchy Integral Formula. Computation of real integrals.
Mar. 4: Consequences of Cauchy Integral Formula. Functions with finite order singularity.
Mar. 7: Consequences of Cauchy Integral Formula. Liouville Theorem: bounded entire functions are constants.
March 9: Maximum Modulus Principle.
March 11: Argument Principle,Rouche's Theorem (Lecture Note 6-5).
March 14: Nyquist criterion, applications to ODE.
March 16: Nyquist criterion. Introduction to Laurant series .(Lecture Note 7).
March 18:Introduction to Laurant series .(Lecture Note 7). Classification of singularities and computations of residues. See explanations of the notation O(...) here Notation $O(z^m)$
March 21: Lecture Note 7. Cauchy Residue Theorem. Computation of residues and contour integrals.
March 23: Applications of Cauchy Residue Theorem. Lecture Note 8, page 1-3.
March 25: Midterm 2
March 28: Type I, Type II real integrals.
March 30: Type III real integrals.
April 1: Type IV and Type V integrals. Integrals involving Multi-valued functions.
April 4: Type V integrals. Finish lecture 8.
April 6 (last class): Fourier transforms and inverse Fourier transforms. Two Properties. Applications to ODE and PDE.
APril 8: Review
Announcements For MATH 305
Back to My Home Page
Feb. 18: Cauchy-Coursat Theorem. simply-connected domains. Path independence and deformation of path.
Feb. 21-25: Spring Break
Feb. 28: Path Independence. Cauchy Integral Formula. Examples.
Mar. 2: Applications of Cauchy Integral Formula. Computation of real integrals.
Mar. 4: Consequences of Cauchy Integral Formula. Functions with finite order singularity.
Mar. 7: Consequences of Cauchy Integral Formula. Liouville Theorem: bounded entire functions are constants.
March 9: Maximum Modulus Principle.
March 11: Argument Principle,Rouche's Theorem (Lecture Note 6-5).
March 14: Nyquist criterion, applications to ODE.
March 16: Nyquist criterion. Introduction to Laurant series .(Lecture Note 7).
March 18:Introduction to Laurant series .(Lecture Note 7). Classification of singularities and computations of residues. See explanations of the notation O(...) here Notation $O(z^m)$
March 21: Lecture Note 7. Cauchy Residue Theorem. Computation of residues and contour integrals.
March 23: Applications of Cauchy Residue Theorem. Lecture Note 8, page 1-3.
March 25: Midterm 2
March 28: Type I, Type II real integrals.
March 30: Type III real integrals.
April 1: Type IV and Type V integrals. Integrals involving Multi-valued functions.
April 4: Type V integrals. Finish lecture 8.
April 6 (last class): Fourier transforms and inverse Fourier transforms. Two Properties. Applications to ODE and PDE.
APril 8: Review
Announcements For MATH 305
Back to My Home Page
Feb. 21-25: Spring Break
Feb. 28: Path Independence. Cauchy Integral Formula. Examples.
Mar. 2: Applications of Cauchy Integral Formula. Computation of real integrals.
Mar. 4: Consequences of Cauchy Integral Formula. Functions with finite order singularity.
Mar. 7: Consequences of Cauchy Integral Formula. Liouville Theorem: bounded entire functions are constants.
March 9: Maximum Modulus Principle.
March 11: Argument Principle,Rouche's Theorem (Lecture Note 6-5).
March 14: Nyquist criterion, applications to ODE.
March 16: Nyquist criterion. Introduction to Laurant series .(Lecture Note 7).
March 18:Introduction to Laurant series .(Lecture Note 7). Classification of singularities and computations of residues. See explanations of the notation O(...) here Notation $O(z^m)$
March 21: Lecture Note 7. Cauchy Residue Theorem. Computation of residues and contour integrals.
March 23: Applications of Cauchy Residue Theorem. Lecture Note 8, page 1-3.
March 25: Midterm 2
March 28: Type I, Type II real integrals.
March 30: Type III real integrals.
April 1: Type IV and Type V integrals. Integrals involving Multi-valued functions.
April 4: Type V integrals. Finish lecture 8.
April 6 (last class): Fourier transforms and inverse Fourier transforms. Two Properties. Applications to ODE and PDE.
APril 8: Review
Announcements For MATH 305
Back to My Home Page
Mar. 2: Applications of Cauchy Integral Formula. Computation of real integrals.
Mar. 4: Consequences of Cauchy Integral Formula. Functions with finite order singularity.
Mar. 7: Consequences of Cauchy Integral Formula. Liouville Theorem: bounded entire functions are constants.
March 9: Maximum Modulus Principle.
March 11: Argument Principle,Rouche's Theorem (Lecture Note 6-5).
March 14: Nyquist criterion, applications to ODE.
March 16: Nyquist criterion. Introduction to Laurant series .(Lecture Note 7).
March 18:Introduction to Laurant series .(Lecture Note 7). Classification of singularities and computations of residues. See explanations of the notation O(...) here Notation $O(z^m)$
March 21: Lecture Note 7. Cauchy Residue Theorem. Computation of residues and contour integrals.
March 23: Applications of Cauchy Residue Theorem. Lecture Note 8, page 1-3.
March 25: Midterm 2
March 28: Type I, Type II real integrals.
March 30: Type III real integrals.
April 1: Type IV and Type V integrals. Integrals involving Multi-valued functions.
April 4: Type V integrals. Finish lecture 8.
April 6 (last class): Fourier transforms and inverse Fourier transforms. Two Properties. Applications to ODE and PDE.
APril 8: Review
Announcements For MATH 305
Back to My Home Page
Mar. 4: Consequences of Cauchy Integral Formula. Functions with finite order singularity.
Mar. 7: Consequences of Cauchy Integral Formula. Liouville Theorem: bounded entire functions are constants.
March 9: Maximum Modulus Principle.
March 11: Argument Principle,Rouche's Theorem (Lecture Note 6-5).
March 14: Nyquist criterion, applications to ODE.
March 16: Nyquist criterion. Introduction to Laurant series .(Lecture Note 7).
March 18:Introduction to Laurant series .(Lecture Note 7). Classification of singularities and computations of residues. See explanations of the notation O(...) here Notation $O(z^m)$
March 21: Lecture Note 7. Cauchy Residue Theorem. Computation of residues and contour integrals.
March 23: Applications of Cauchy Residue Theorem. Lecture Note 8, page 1-3.
March 25: Midterm 2
March 28: Type I, Type II real integrals.
March 30: Type III real integrals.
April 1: Type IV and Type V integrals. Integrals involving Multi-valued functions.
April 4: Type V integrals. Finish lecture 8.
April 6 (last class): Fourier transforms and inverse Fourier transforms. Two Properties. Applications to ODE and PDE.
APril 8: Review
Announcements For MATH 305
Back to My Home Page
Mar. 7: Consequences of Cauchy Integral Formula. Liouville Theorem: bounded entire functions are constants.
March 9: Maximum Modulus Principle.
March 11: Argument Principle,Rouche's Theorem (Lecture Note 6-5).
March 14: Nyquist criterion, applications to ODE.
March 16: Nyquist criterion. Introduction to Laurant series .(Lecture Note 7).
March 18:Introduction to Laurant series .(Lecture Note 7). Classification of singularities and computations of residues. See explanations of the notation O(...) here Notation $O(z^m)$
March 21: Lecture Note 7. Cauchy Residue Theorem. Computation of residues and contour integrals.
March 23: Applications of Cauchy Residue Theorem. Lecture Note 8, page 1-3.
March 25: Midterm 2
March 28: Type I, Type II real integrals.
March 30: Type III real integrals.
April 1: Type IV and Type V integrals. Integrals involving Multi-valued functions.
April 4: Type V integrals. Finish lecture 8.
April 6 (last class): Fourier transforms and inverse Fourier transforms. Two Properties. Applications to ODE and PDE.
APril 8: Review
Announcements For MATH 305
Back to My Home Page
March 9: Maximum Modulus Principle.
March 11: Argument Principle,Rouche's Theorem (Lecture Note 6-5).
March 14: Nyquist criterion, applications to ODE.
March 16: Nyquist criterion. Introduction to Laurant series .(Lecture Note 7).
March 18:Introduction to Laurant series .(Lecture Note 7). Classification of singularities and computations of residues. See explanations of the notation O(...) here Notation $O(z^m)$
March 21: Lecture Note 7. Cauchy Residue Theorem. Computation of residues and contour integrals.
March 23: Applications of Cauchy Residue Theorem. Lecture Note 8, page 1-3.
March 25: Midterm 2
March 28: Type I, Type II real integrals.
March 30: Type III real integrals.
April 1: Type IV and Type V integrals. Integrals involving Multi-valued functions.
April 4: Type V integrals. Finish lecture 8.
April 6 (last class): Fourier transforms and inverse Fourier transforms. Two Properties. Applications to ODE and PDE.
APril 8: Review
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March 11: Argument Principle,Rouche's Theorem (Lecture Note 6-5).
March 14: Nyquist criterion, applications to ODE.
March 16: Nyquist criterion. Introduction to Laurant series .(Lecture Note 7).
March 18:Introduction to Laurant series .(Lecture Note 7). Classification of singularities and computations of residues. See explanations of the notation O(...) here Notation $O(z^m)$
March 21: Lecture Note 7. Cauchy Residue Theorem. Computation of residues and contour integrals.
March 23: Applications of Cauchy Residue Theorem. Lecture Note 8, page 1-3.
March 25: Midterm 2
March 28: Type I, Type II real integrals.
March 30: Type III real integrals.
April 1: Type IV and Type V integrals. Integrals involving Multi-valued functions.
April 4: Type V integrals. Finish lecture 8.
April 6 (last class): Fourier transforms and inverse Fourier transforms. Two Properties. Applications to ODE and PDE.
APril 8: Review
Announcements For MATH 305
Back to My Home Page
March 14: Nyquist criterion, applications to ODE.
March 16: Nyquist criterion. Introduction to Laurant series .(Lecture Note 7).
March 18:Introduction to Laurant series .(Lecture Note 7). Classification of singularities and computations of residues. See explanations of the notation O(...) here Notation $O(z^m)$
March 21: Lecture Note 7. Cauchy Residue Theorem. Computation of residues and contour integrals.
March 23: Applications of Cauchy Residue Theorem. Lecture Note 8, page 1-3.
March 25: Midterm 2
March 28: Type I, Type II real integrals.
March 30: Type III real integrals.
April 1: Type IV and Type V integrals. Integrals involving Multi-valued functions.
April 4: Type V integrals. Finish lecture 8.
April 6 (last class): Fourier transforms and inverse Fourier transforms. Two Properties. Applications to ODE and PDE.
APril 8: Review
Announcements For MATH 305
Back to My Home Page
March 16: Nyquist criterion. Introduction to Laurant series .(Lecture Note 7).
March 18:Introduction to Laurant series .(Lecture Note 7). Classification of singularities and computations of residues. See explanations of the notation O(...) here Notation $O(z^m)$
March 21: Lecture Note 7. Cauchy Residue Theorem. Computation of residues and contour integrals.
March 23: Applications of Cauchy Residue Theorem. Lecture Note 8, page 1-3.
March 25: Midterm 2
March 28: Type I, Type II real integrals.
March 30: Type III real integrals.
April 1: Type IV and Type V integrals. Integrals involving Multi-valued functions.
April 4: Type V integrals. Finish lecture 8.
April 6 (last class): Fourier transforms and inverse Fourier transforms. Two Properties. Applications to ODE and PDE.
APril 8: Review
Announcements For MATH 305
Back to My Home Page
March 18:Introduction to Laurant series .(Lecture Note 7). Classification of singularities and computations of residues. See explanations of the notation O(...) here Notation $O(z^m)$
March 21: Lecture Note 7. Cauchy Residue Theorem. Computation of residues and contour integrals.
March 23: Applications of Cauchy Residue Theorem. Lecture Note 8, page 1-3.
March 25: Midterm 2
March 28: Type I, Type II real integrals.
March 30: Type III real integrals.
April 1: Type IV and Type V integrals. Integrals involving Multi-valued functions.
April 4: Type V integrals. Finish lecture 8.
April 6 (last class): Fourier transforms and inverse Fourier transforms. Two Properties. Applications to ODE and PDE.
APril 8: Review
Announcements For MATH 305
Back to My Home Page
March 21: Lecture Note 7. Cauchy Residue Theorem. Computation of residues and contour integrals.
March 23: Applications of Cauchy Residue Theorem. Lecture Note 8, page 1-3.
March 25: Midterm 2
March 28: Type I, Type II real integrals.
March 30: Type III real integrals.
April 1: Type IV and Type V integrals. Integrals involving Multi-valued functions.
April 4: Type V integrals. Finish lecture 8.
April 6 (last class): Fourier transforms and inverse Fourier transforms. Two Properties. Applications to ODE and PDE.
APril 8: Review
Announcements For MATH 305
Back to My Home Page
March 23: Applications of Cauchy Residue Theorem. Lecture Note 8, page 1-3.
March 25: Midterm 2
March 28: Type I, Type II real integrals.
March 30: Type III real integrals.
April 1: Type IV and Type V integrals. Integrals involving Multi-valued functions.
April 4: Type V integrals. Finish lecture 8.
April 6 (last class): Fourier transforms and inverse Fourier transforms. Two Properties. Applications to ODE and PDE.
APril 8: Review
Announcements For MATH 305
Back to My Home Page
March 25: Midterm 2
March 28: Type I, Type II real integrals.
March 30: Type III real integrals.
April 1: Type IV and Type V integrals. Integrals involving Multi-valued functions.
April 4: Type V integrals. Finish lecture 8.
April 6 (last class): Fourier transforms and inverse Fourier transforms. Two Properties. Applications to ODE and PDE.
APril 8: Review
Announcements For MATH 305
Back to My Home Page
March 28: Type I, Type II real integrals.
March 30: Type III real integrals.
April 1: Type IV and Type V integrals. Integrals involving Multi-valued functions.
April 4: Type V integrals. Finish lecture 8.
April 6 (last class): Fourier transforms and inverse Fourier transforms. Two Properties. Applications to ODE and PDE.
APril 8: Review
Announcements For MATH 305
Back to My Home Page
March 30: Type III real integrals.
April 1: Type IV and Type V integrals. Integrals involving Multi-valued functions.
April 4: Type V integrals. Finish lecture 8.
April 6 (last class): Fourier transforms and inverse Fourier transforms. Two Properties. Applications to ODE and PDE.
APril 8: Review
Announcements For MATH 305
Back to My Home Page
April 1: Type IV and Type V integrals. Integrals involving Multi-valued functions.
April 4: Type V integrals. Finish lecture 8.
April 6 (last class): Fourier transforms and inverse Fourier transforms. Two Properties. Applications to ODE and PDE.
APril 8: Review
Announcements For MATH 305
Back to My Home Page
April 4: Type V integrals. Finish lecture 8.
April 6 (last class): Fourier transforms and inverse Fourier transforms. Two Properties. Applications to ODE and PDE.
APril 8: Review
Announcements For MATH 305
Back to My Home Page
April 6 (last class): Fourier transforms and inverse Fourier transforms. Two Properties. Applications to ODE and PDE.
APril 8: Review
Announcements For MATH 305
Back to My Home Page
APril 8: Review
Announcements For MATH 305
Back to My Home Page