91Ó°ÊÓ

Discuss the type of singularity (removable, pole and order, essential, branch, cluster, natural barrier, etc.); if the type is a pole give the strength of the pole, and give the nature (isolated or not) of all singular points associated with the following functions. Include the point at infinity. (a) \(\frac{e^{z^{2}}-1}{z^{2}}\) (b) \(\frac{e^{2 z}-1}{z^{2}}\) (c) \(e^{\tan z}\) (d) \(\frac{z^{3}}{z^{2}+z+1}\) (e) \(\frac{z^{1 / 3}-1}{z-1}\) (f) \(\log \left(1+z^{1 / 2}\right)\) (g) \(f(z)=\left\\{\begin{array}{cc}z^{2} & |z| \leq 1 \\ 1 / z^{2}|z|>1\end{array}\right.\) (h) \(f(z)=\sum_{n=1}^{\infty} \frac{z^{n !}}{n !}\) (i) \(\operatorname{sech} z\) (j) \(\operatorname{coth} 1 / z\)

Expand the function \(f(z)=1 /\left(1+z^{2}\right)\) in (a) a Taylor series for \(|z|<1\) (b) a Laurent series for \(|z|>1\)

Show that if \(f(z)\) is meromorphic in the finite \(z\) plane, then \(f(z)\) must be the ratio of two entire functions.

Use the Taylor series representation of \(1 /(1-z)\) around \(z=0\), for \(|z|<1\), to deduce the series representation of \(1 /(1-z)^{2}, 1 /(1-z)^{3}, \ldots\), \(1 /(1-z)^{m \cdot} .\)

Use the binomial expansion and Cauchy's Integral Theorem to evaluate $$ \oint_{C}(z+1 / z)^{2 n} \frac{d z}{z} $$ where \(C\) is the unit circle centered at the origin. Recall the binomial expansion $$ (a+b)^{n}=a^{n}+n a^{n-1} b+\cdots=\sum_{k=0}^{n}\left(\begin{array}{l} n \\ k \end{array}\right) a^{n-k} b^{k} $$ where $$ \left(\begin{array}{l} n \\ k \end{array}\right) \equiv \frac{n !}{k !(n-k) !} $$ Use this result to establish the following real integral formula: $$ \frac{1}{2 \pi} \int_{0}^{2 \pi}(\cos \theta)^{2 n} d \theta=\frac{(2 n) !}{4^{n}(n !)^{2}} $$