# Mathematics Department The University of Georgia Math 8150 Homework Assignment 3

Due 10 pm on 2/9/2021. Late homework will not be accepted

1. Assume real value functions u, v of two variables have continuous partial derivatives at $( x _ { 0 } , y _ { 0 } )$ . Show that $f = u + i v$ has derivative $f ^ { \prime } ( z _ { 0 } )$ at $z _ { 0 } = x _ { 0 } + i y _ { 0 }$ if and only if

$$
\operatorname * { l i m } _ { r  0 } \frac { 1 } { \pi r ^ { 2 } } \int _ { | z - z _ { 0 } | = r } f ( z ) d z = 0 \ .
$$

2. (Cauchy’s formula for “exterior” region) Let γ be piecewise smooth simple closed curve with interior $\Omega _ { 1 }$ and exterior $\Omega _ { 2 }$ . Assume $f ^ { \prime } ( z )$ exists in an open set containing $\gamma$ and $\Omega _ { 2 }$ and $\begin{array} { r } { \operatorname* { l i m } _ { z \to \infty } f ( z ) = A } \end{array}$ . Show that

$$
{ \frac { 1 } { 2 \pi i } } \int _ { \gamma } { \frac { f ( \xi ) } { \xi - z } } d \xi = { \left\{ \begin{array} { l l } { A , } & { { \mathrm { i f ~ } } z \in \Omega _ { 1 } , } \\ { - f ( z ) + A , } & { { \mathrm { i f ~ } } z \in \Omega _ { 2 } } \end{array} \right. }
$$

3. Let $f ( z )$ be bounded and analytic in $\mathbb { C } .$ . Let $a \neq b$ be any fixed complex numbers. Show that the following limit exists

$$
\operatorname* { l i m } _ { R \to \infty } \int _ { | z | = R } { \frac { f ( z ) } { ( z - a ) ( z - b ) } } d z .
$$

Use this to show that $f ( z )$ must be a constant (Liouville’s theorem).

4. Suppose that $f ( z )$ is entire and lim $_ { z \to \infty } f ( z ) / z = 0$ . Show that $f ( z )$ is a constant.

5. Let f be analytic on a domain D and let $\gamma$ be a closed curve in $D .$ . For any $z _ { 0 }$ in $D$ not on $\gamma ,$ show that

$$
\int _ { \gamma } \frac { f ^ { \prime } ( z ) } { ( z - z _ { 0 } ) } d z = \int _ { \gamma } \frac { f ( z ) } { ( z - z _ { 0 } ) ^ { 2 } } d z
$$

Give a generalization of this result.

6. Compute $\int _ { | z | = 1 } \left( z + { \frac { 1 } { z } } \right) ^ { 2 n } { \frac { d z } { z } }$ and use it to show that

$$
\int _ { 0 } ^ { 2 \pi } \cos ^ { 2 n } \theta d \theta = 2 \pi { \frac { 1 \cdot 3 \cdot 5 \cdot \cdot \cdot ( 2 n - 1 ) } { 2 \cdot 4 \cdot 6 \cdot \cdot \cdot ( 2 n ) } } .
$$

7. Prove that

$$
\int _ { 0 } ^ { \infty } \sin ( x ^ { 2 } ) d x = \int _ { 0 } ^ { \infty } \cos ( x ^ { 2 } ) d x = { \frac { \sqrt { 2 \pi } } { 4 } } .
$$

These are the Fresnel integrals. Here $\int _ { 0 } ^ { \infty }$ is interpreted as $\operatorname* { l i m } _ { R \to \infty } \int _ { 0 } ^ { R }$

[For a Hint, see textbook by Stein and Shakarchi p64, #1]

8. Show that $\int _ { 0 } ^ { \infty } { \frac { \sin { x } } { x } } d x = { \frac { \pi } { 2 } }$

[For a Hint, see textbook by Stein and Shakarchi p64, #2]

9. Goursat’s theorem under the additional hypothesis that $f ^ { \prime }$ is continuous:

Suppose $f ^ { \prime } ( z )$ is continuous on a region Ω, and that the triangle T along with its interior is in Ω. Apply Green’s theorem to show that

$$
\int _ { T } f ( z ) d z = 0 .
$$

[For a Hint, see textbook by Stein and Shakarchi p65, #5]

10. Suppose $f ^ { \prime } ( z )$ exists in a region Ω except possibly at point w in Ω. Let $T \subset \Omega$ b e a triangle whose interior is also contained in Ω and contains w. Show that

$$
\int _ { T } f ( z ) d z = 0 .
$$

11. Let f be a holomorphic function on the strip $- 1 < y < 1 , x \in \mathbb { R }$ such that

$$
| f ( z ) | \leq A ( 1 + | z | ) ^ { \eta } , \quad \eta \mathrm { ~ i s ~ a ~ f x e d ~ r e a l ~ n u m b e r }
$$

for all z in the strip. Show that for each integer $n \geq 0$ , there exists $A _ { n }$ such that

$$
| f ^ { ( n ) } ( x ) | \leq A _ { n } ( 1 + | x | ) ^ { \eta } , \quad { \mathrm { ~ f o r ~ a l l ~ } } x \in \mathbb { R } .
$$

[Hint: Use Cauchy inequalities]