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{\bf Math 122-A \hskip.1in   Winter 2022\hskip.1in HOMEWORK\# 7-8 (due March. 9th)
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(1) Let $D_{1}\left(z_{0}\right)=\left\{z \in \mathbb{C}:\left|z-z_{0}\right|<1\right\} .$ Let $f, g: D_{1}\left(z_{0}\right) \rightarrow \mathbb{C}$ be two analytic functions on $D_{1}\left(z_{0}\right)$. Prove that if
$$
f^{(n)}\left(z_{0}\right)=g^{(n)}\left(z_{0}\right), \quad n=0,1,2,3, \ldots \ldots .
$$
then $f(z)=g(z), \forall z \in D_{1}\left(z_{0}\right)$.


(2) Let $D_{1}\left(z_{0}\right)=\left\{z \in \mathbb{C}:\left|z-z_{0}\right|<1\right\}$. Let $f: D_{1}\left(z_{0}\right) \rightarrow \mathbb{C}$ be an analytic function on $D_{1}\left(z_{0}\right)$ such that it has a zero of order $N \in \mathbb{N}$ at $z_{0}$, i.e.
$$
f\left(z_{0}\right)=f^{\prime}\left(z_{0}\right)=\ldots .=f^{N-1}\left(z_{0}\right)=0, \quad f^{n}\left(z_{0}\right) \neq 0 .
$$
(i) Prove that exists $g: D_{1}\left(z_{0}\right) \rightarrow \mathbb{C}$ analytic on $D_{1}\left(z_{0}\right)$ with $g\left(z_{0}\right) \neq 0$ and $f(z)=\left(z-z_{0}\right)^{N} g(z)$ .

(ii) There exists $\delta>0$ such that if $0<\left|z-z_{0}\right|<\delta$ such that $f(z) \neq 0$.
(The zeros of a non-trivial analytic function are isolated)

(3) Let $f(z)=\sin (\pi / z)$. Thus, $f(1 / n)=0$. Does this contradict the result in (2)?

(4) Find the order of each of the zeros of the given functions:

(a) $\left(z^{2}-4 z+4\right)^{2}, \quad$ (b) $z^{2}(1-\cos (z)), \quad$ (c) $e^{2 z}-3 e^{z}-4$

(5) Locate the isolated singularity of the given function and tell whether is removable
singularity, a pole or an essential singularity. $$ \begin{array}{llll}(a) \frac{e^{z}-1}{z}, & (b) \frac{z^{2}}{\sin (z)}, & (c) \frac{e^{z}-1}{e^{2 z}-1}, & (d) \frac{z^{4}-2 z^{2}+1}{(z-1)^{2}}\end{array} $$

(6) Find the Laurent series for a given function about the point $z=0$ and find the residue at that point
(a) $\frac{e^{z}-1}{z}, \quad(b) \frac{z}{(\sin (z))^{2}}, \quad(c) \frac{1}{e^{z}-1}, \quad(d) \frac{1}{1-\cos (z)} .$
In (c) and (d) compute only three terms of the Laurent series.

(7) Find the residue of $f(z)=1 /\left(1+z^{n}\right)$ at the point $z_{0}=e^{i \pi / n}$.

(8) Calculate :
(a) $\int_{-\infty}^{\infty} \frac{x^{2}}{\left(1+x^{2}\right)\left(4+x^{2}\right)} d x$

(b) $\int_{-\infty}^{\infty} \frac{d x}{\left(1+x^{2}\right)^{2}}(=\pi / 2)$,

(c) $\int_{-\infty}^{\infty} \frac{x \sin (a x)}{x^{2}+b^{2}} d x\left(=\pi e^{-a b}\right)$,

(d) $\int_{-\infty}^{\infty} \frac{\sin (x)}{x} d x(=\pi)$,

(e) $\int_{0}^{2 \pi} \frac{d t}{2+\cos ^{2}(t)}$



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