Assignment 8: Contour Integration

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import mmf_setup;mmf_setup.nbinit()
import logging;logging.getLogger('matplotlib').setLevel(logging.CRITICAL)
%matplotlib inline
import numpy as np, matplotlib.pyplot as plt

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Assignment 8: Contour Integration#

Due Mon 27 Oct 2025 at the start of class

1. Residues#

Determine the types of singularities in each of the following and compute the residue. (Assume \(a>0\) is real and positive.)

\[\begin{align*} A(z) &= \frac{1}{z^2+a^2} & B(z) &= \frac{1}{(z^2+a^2)^2}\\ C(z) &= \frac{z^2}{z^2+a^2} & D(z) &= \frac{\sin(1/z)}{z^2+a^2}\\ E(z) &= \frac{ze^{+\I z}}{z^2+a^2} & F(z) &= \frac{ze^{+\I z}}{z^2-a^2}\\ G(z) &= \frac{e^{+\I z}}{z^2-a^2} & H(z) &= \frac{z^{-k}}{z+1}, \quad 0 < k < 1 \end{align*}\]

2. Contour Integrals#

Evaluate the following integrals using contours:

\[\begin{align*} I_1 &= \int_0^{2\pi}\frac{\cos 3\theta}{5-4\cos\theta}\d{\theta}, & I_2 &= \int_0^{\infty}\frac{x\sin x}{x^2+1}\d{x}, & I_3 &= \int_0^{\infty}\frac{x^p \ln x}{x^2 + 1}\d{x}, \quad 0 < p < 1. \end{align*}\]

3. Möbius Transformations#

What part of the \(z\)-plane corresponds to the interior of the unit circle for each of the following two conformal maps:

\[\begin{align*} w &= \frac{z-1}{z+1} & w &= \frac{z-\I}{z+\I}. \end{align*}\]

Carefully sketch the behaviour of these maps in a manner similar to that shown in [Needham, 2023] or in my notes. (Choose lines or grids as needed to clearly express the behaviour of the transformation, but be able to sketch these by hand as you would need to in an exam setting.)

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