# Mathematics Department The University of Georgia Math 8150 Homework Assignment 2

Due 10 pm on 1/28/2021. Late homework will not be accepted

1. Suppose $U ( z )$ has continuous second order partial derivatives and $z = f ( \zeta ) , \zeta = \xi + i \eta$ is a holomorphic function. Show that

$$
\frac { \partial ^ { 2 } U } { \partial \xi ^ { 2 } } + \frac { \partial ^ { 2 } U } { \partial \eta ^ { 2 } } = | f ^ { \prime } ( \zeta ) | ^ { 2 } \left[ \frac { \partial ^ { 2 } U } { \partial x ^ { 2 } } + \frac { \partial ^ { 2 } U } { \partial y ^ { 2 } } \right] .
$$

2. Show that $U ( x ^ { 2 } - y ^ { 2 } , 2 x y )$ is harmonic if and only if $U ( x , y )$ is.

3. Let $a _ { n } \neq 0$ and assume that lim ${ \frac { \left| a _ { n + 1 } \right| } { \left| a _ { n } \right| } } = L .$ . Show that lim ${ \sqrt [ n ] { \left| a _ { n } \right| } } = L$ . In particular, n→∞ n→∞ this shows that when applicable, the ratio test can be used to calculate the radius of convergence of a power series.

4. Let f be a power series centered at the origin. Prove that f has a power series expansion around any point in its disc of convergence.

5. Prove the following:

(a) The power series $\sum _ { n = 1 } ^ { \infty } n z ^ { n }$ does not converge at any point of the unit circle.

(b) The power series $\sum _ { n = 1 } ^ { \infty } { \frac { z ^ { n } } { n ^ { 2 } } }$ converges at every point of the unit circle.

(c) The power series $\sum _ { n = 1 } ^ { \infty } { \frac { z ^ { n } } { n } }$ converges at every point of the unit circle except at $z = 1$

6. Show that the series

$$
\sum _ { n = 1 } ^ { \infty } { \frac { 1 } { n ^ { z } } }
$$

defines a continuous function for $\mathrm { R e } ( z ) > 1$ . (This is the Riemann ζ function and you will see later that it is analytic in the above region-not just being continuous thereand it can be extended to the complex plane with only 1 removed. You do not need to prove the latter assertions.) [Hint: Show that the series converges uniformly for $\mathrm { R e } ( z ) \geq 1 + \delta$ , where $\delta > 0$ is any positive number.]

7. Show that the series

$$
\sum _ { n = 1 } ^ { \infty } { \frac { ( - 1 ) ^ { n } } { n } } z ^ { n ( n + 1 ) }
$$

has radius of convergence 1. Examine convergence at $z = 1 , - 1$ and i.

8. It was defined that $e ^ { z } : = \sum _ { n = 0 } ^ { \infty } \frac { z ^ { n } } { n ! } .$ . Show that $e ^ { z } = \operatorname* { l i m } _ { n \to \infty } \left( 1 + { \frac { z } { n } } \right) ^ { n }$

9. (a) Express sin z in the form $u + i v$ . Do the same for cos z.

(b) Show that there exists a sequence $z _ { n }$ such that sin $z _ { n } \to \infty$ . Do the same for cos z.

(This shows that unlike their real counter parts, sin z and cos z are unbounded.)

10. Prove that

$$
\mathrm { ( a ) } \operatorname* { l i m } _ { z \to ( 2 k + 1 ) \pi / 2 } \tan z = \infty , \mathrm { ( b ) } \operatorname* { l i m } _ { z \to ( 2 k + 1 ) \pi / 2 } [ z - ( 2 k + 1 ) \pi / 2 ] \tan z = - 1 .
$$

[Hint: The proof of (b) is very short if done the “right” way.]

11. Show that if $\vert \alpha \vert < r < \vert \beta \vert$ , then

$$
\int _ { \gamma } { \frac { 1 } { ( z - \alpha ) ( z - \beta ) } } = { \frac { 2 \pi i } { \alpha - \beta } }
$$

where γ denotes the circle centered at the origin, of radius r, with positive orientation.

12. Assume f is continuous in the region: $x \ge x _ { 0 } , 0 \le y \le b$ and the limit

$$
\operatorname* { l i m } _ { x \to + \infty } f ( x + i y ) = A
$$

exists uniformly with respect to y (independent of y). Show that

$$
\operatorname* { l i m } _ { x  + \infty } \int _ { \gamma _ { x } } f ( z ) d z = i A b ,
$$

where $\gamma _ { x } : = \{ z \mid z = x + i t , 0 \leq t \leq b \}$