Assignment 7: Complex Analysis

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import numpy as np, matplotlib.pyplot as plt

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Assignment 7: Complex Analysis#

Due Mon 20 Oct 2025 at the start of class

1. Complex Numbers#

Find the Cartesian form for all values of

  1. \((-8)^{1/3}\).

  2. \(\I^{1/4}\).

  3. \(e^{\I \pi/4}\).

  4. \(\cos(1+\pi \I)\).

Sketch their locations in the complex plane and circle (or put a box around) the principal value returned by the corresponding function in Python.

2. Analytic Functions#

Find the analytic function \(f(z) = u(x,y) + \I v(x,y)\)

  1. if \(u(x, y) = x^3 - 3xy^2\),

  2. if \(v(x, y) = e^{-y}\sin x\).

Express your answers in terms of \(z\). (I.e. show that \(f(z)\) has no dependence on \(z^*\).)

Based on your work, do you think we can choose any \(u(x, y)\) and expect to be able to find a \(v(x, y)\) that makes \(f(z)\) analytic? If not, give an example of a restrictive condition \(u(x, y)\) must satisfy.

3. A Simple Contour Integral#

Evaluate the following contour integral

\[\begin{gather*} \oint_{C} \frac{\d{z}}{z^2-1} \end{gather*}\]

counter-clockwise over the contour \(C\) where \(C\) is the circle \(\abs{z-1} = 1\).

4. A Box Contour#

Evaluate the following contour integral

\[\begin{gather*} \oint_{C} \frac{e^{\I z}}{z^3}\d{z} \end{gather*}\]

counter-clockwise over a square with sides of length \(a > 1\) centered at \(z=0\).

5. Laurent Series#

Obtain the Laurent expansion of the following about \(z=1\):

\[\begin{gather*} \frac{ze^{z}}{z-1}. \end{gather*}\]

6. Check Your Results#

Check your results numerically. For example:

    1. Evaluate the various quantities with python.

    1. Numerically check your expressions to check your algebra.

    1. and 4. Explicitly formulate the contour integrals, and then integrate with quad.

    1. Numerically sum the series close to \(z=1\) to make sure you get the correct answer.

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