# 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.

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# Lecture Notes For MATH305

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# Downloads For MATH305

# Updates For MATH 305

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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### Feb. 10: Midterm 1

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

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

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

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

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

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### March 6: Maximum Modulus Principle.

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### March 8: Argument Principle, Nyquist Criterion.

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

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

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

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

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

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### March 22: Type I, Type II real integrals.

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### March 24: Midterm 2

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### March 27: Type III real integrals.

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

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### April 3: Type V integrals. Finish lecture 8.

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

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# Announcements For MATH 305

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

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

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

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

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### New deadline for HW9: March 22.

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

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

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

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