MATH305-201 :       Applied Analysis and Complex Variables   (2nd term 2016/2017)

Lecture   I: Monday 12noon--1:00pm, BUCH-A201.

Lecture   II: Wednesday 12noon--1:00pm, BUCH-A201.

Lecture   III: Friday 12noon--1:00pm, BUCH-A201.

Office Hours: Every Monday, Wednesday, Friday, 4:30pm-5:30pm, LSK 303B.

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 Residue Calculus

Lecture Notes 8 Lecture Notes 8-Notes on Integration by Residue Calculus

Lecture Notes 8-5 Lecture Notes 8-5: A Summary

Lecture Notes 9 Lecture Notes 9- Notes on Laurent Series, Singularities and Residue Calculus

Lecture Notes 10 Lecture Notes 10- Notes on Fourier Transforms and Applications

Downloads For MATH305

Download 1: Syllabus

Download 2: HW1 (Due Date: Jan. 13, by 5:30pm)

Download 3: Solutions to HW1

Download 4: HW2 (Due Date: Jan. 23, by 5:30pm)

Download 5: HW3 (Due Date: Jan. 30, by 5:30pm)

Download 6: Solutions to HW2

Download 7: HW4 (Due Date: Feb. 6, by 5:30pm)

Download 8: Solutions to HW3

Download 9: Past Midterm 1-1

Download 10: Past Midterm 1-2

Download 11: Past Midterm 1-3

Download 12: HW5 (Due Date: Feb. 15, by 5:30pm)

Download 13: Solutions to HW4

Download 14: Midterm One and Solutions

Download 15: Solutions to HW5

Download 16: HW6 (Due: Feb. 27)

Download 17: Solutions to HW6

Download 18: HW7 (Due: March 6)

Download 19: Solutions to HW7

Download 20: HW8 (Due: March 13)

Download 21: Solutions to HW8

Download 22: HW9 (Due: March 22)

Download 23: Past Midterm 2-1

Download 24: Past Midterm 2-2

Download 25: Past Midterm 2-3

Download 26: Past Sample Problems

Download 27: Solutions to HW9

Download 28: Solutions to Midterm Examination 2

Download 29: HW10 (Due: April 7)

Download 30: Solutions to HW10

Download 31: Past Review Problems

Download 32: Solutions to Past Review Problems

Download 33: Past Final Exam

Updates For MATH 305

Jan. 4: Fundamentals of complex variable. Euler's formula. Polar coordinate. Principal value of argument Arg (z).

Jan. 6: Arg (z) and arg (z). De Moivre's formula. Roots of unit. Roots of a complex variable.

Jan. 9: Complex exponential. Sets in the complex plane. Functions of complex variables.

Jan. 11: Functions of complex variables. Image under linear and Mobius map $ w=(a+bz)/ (c+dz)$.

Jan. 13: Image under $w=z^2$. Continuous, differentiable, analytic. Cauchy-Riemann equation.

Jan. 16: Consequences of Cauchy-Riemann equation. Harmonic Functions. Conformal Mapping. Level Sets. More notes can be found here. Lecture Notes on Conformal Mappings

Jan. 18: Laplace under analytical mappings. $\partial_{\bar{z}} f(z)=0$. Conformal Mappings. Elementary Functions.

Jan. 20: Elementary functions $ e^z$ and $ \sin (z)$. Images under $ e^z$ and $ sin (z)$.

Jan. 23: Properties of $\sin z$ and $ sinh (z)$. Introduction of $Log (z)$.

Jan. 25: Multi-valued functions. introduction of $log (z)$ and $Log (z)$ and their properties.

Jan. 27: Multi-valued functions. Introduction of $z^\alpha$ and branch cuts.

Jan. 30: Multi-valued functions. Branch cuts for $ (z^2-1)^{\frac{1}{2}}$.

Feb. 01: Branch Cuts for $ (z^3-z)^{1/2}, (z^3-z)^{1/3}, (z^2+1)^{1/2}$.

Feb. 03: Inverse function of sin (z). Solving Laplace equation with Arg (z)

Feb. 06: Complex integrals. Contours (Paths).

Feb. 08: Fundamental Theorem of Calculus in the Complex Case. Examples.

Feb. 10: Midterm 1

Feb. 15: Cauchy-Coursat Theorem. simply-connected domains. Path independence and deformation of path.

Feb. 17: Path Independence. Cauchy Integral Formula. Examples.

Feb. 27: Applications of Cauchy Integral Formula. Computation of real integrals.

Mar. 1: Consequences of Cauchy Integral Formula. Functions with finite order singularity.

Mar. 3: Consequences of Cauchy Integral Formula. Liouville Theorem: bounded entire functions are constants.

March 6: Maximum Modulus Principle.

March 8: Argument Principle, Nyquist Criterion.

March 10: Argument Principle, Nyquist criterion, applications to ODE.

March 13: Rouche's Theorem (Lecture Note 6-5). Classification of Singularities (Lecture Note 7).

March 15: Lecture Note 7. Classification of singularities and computations of residues. See explanations of the notation O(...) here Notation $O(z^m)$

March 17: Lecture Note 7. Cauchy Residue Theorem. Computation of residues and contour integrals.

March 20: Applications of Cauchy Residue Theorem. Lecture Note 8, page 1-3.

March 22: Type I, Type II real integrals.

March 24: Midterm 2

March 27: Type III real integrals.

March 29 Type IV and Type V integrals. Integrals involving Multi-valued functions.

April 3: Type V integrals. Finish lecture 8.

April 5 (last class): Fourier transforms and inverse Fourier transforms. Two Properties. Applications to ODE and PDE.

Announcements For MATH 305

New office hour on Jan. 12:4:30-5:30pm

Coverage for the 1st Midterm: up to Lecture Note 4 on branch cuts and multi-valued functions.

New Office Hours on Tuesday (Feb. 7) and Thursday (Feb. 9): Tuesday, 4pm-5pm; Thursday, 4-5:30pm.

No office hours during the term break (Feb. 20-24).

New deadline for HW9: March 22.

Coverage for the 2nd Midterm: up to page 6 of Lecture Note 8 on real integrals.

New Office Hours on Thursday (March 23): 4-5:30pm.

New Office Hours on Thursday (April 6): 4-5:30pm.

New Office Hours on April 10, 11, 12, 25, 26: 10am-5pm

   Back to My Home Page