Math 136: Complex Variables and Integral Transforms

### Homework 3

Read Sections 2.4, 2.5, 3.1, and 3.2 of Saff and Snider,
and the summary at the end of Chapter 2.

Section 2.7 has a brief introduction to the fascinating Julia sets and Mandelbrot set of complex dynamics, as studied in Math 189: Complex Dynamics.

Do these problems from Saff and Snider:

• Sec. 2.4, The Cauchy-Riemann Equations: 2, 3, 15.
• Sec. 2.5, Harmonic Functions: 1a, 2, 3d, 6, 11.
• Sec. 3.1, Polynomials and Rational Functions: 11d.
• Sec. 3.2, The Exponential, Trigonometric, and Hyperbolic Functions:
5c, f, 11, 14c, 17a, 20.
Hint for 20: You can use the result of Q19 without proof.
• Bonus problems: Sec. 2.4: 6, and Sec. 2.5: 10.

The bonus problems show that when the real and imaginary parts u and v of a function f are expressed in terms of polar coordinates r and t = theta, the Cauchy-Riemann equations become

du/dr = (1/r) dv/dt, and dv/dr = -(1/r) du/dt,

and Laplace's equation becomes

d2u/dr2 + (1/r) du/dr + (1/r2) d2u/dt2 = 0.