Homework 5

Due Monday in class

1. Let $C$ be the contour of $|z|=10$. Compute the following integrals.

(1) $$\oint_C \frac{1}{1+z^2} dz $$

(2) (the result for this one is not zero.) $$\oint_C \frac{z}{1+z^2} dz $$

(3) $$\oint_C \frac{z^2}{1+z^4} dz $$

Apply methods 1,2,3 to the above problems (each method need to be used once)

2. Consider the multivalued function $f(z) = \sqrt{z}$, sketch the motion of the values of $f(z)$ as $z$ moves the following curves

• (1) $z$ move around a radius $1$ circle around $0$, counter clockwise (CCW), $z(t) = e^{i t}$, $t\in [0,2\pi]$.
• (2) $z$ move around a radius $r=1/2$ circle around $1$, CCW, $z(t) = 1 + r e^{i t}$.
• what happens if we change $r$ from $1/2$ to $2$, describe in words.

3. Consider the multivalued function $f(z) = \sqrt{z(z-1)}$, sketch the motion of the values of $f(z)$ as $z$ moves the following curves: $z$ along the circle of radius 10, centered at $0$.