91Ó°ÊÓ

Expand the given function in a Taylor series centered at the indicated point. Give the radius of convergence of each series. \(f(z)=\frac{1}{1+z}, z_{0}=-i\)

Expand \(f(z)=\frac{1}{(z-1)(z-2)}\) in a Laurent series valid for the indicated annular domain. \(0<|z-2|<1\)

In Problems 15-20, determine whether the given geometric series is convergent or divergent. If convergent, find its sum. $$ \sum_{k=1}^{\infty} 4 i\left(\frac{1}{3}\right)^{k-1} $$

Determine the order of the poles for the given function. \(f(z)=\frac{z-1}{(z+1)\left(z^{3}+1\right)}\)

Evaluate the Cauchy principal value of the given improper integral. \(\int_{-\infty}^{\infty} \frac{x}{\left(x^{2}+4\right)^{3}} d x\)

Determine whether the given geometric series is convergent or divergent. If convergent, find its sum. \(\sum_{k=1}^{\infty} 4 i\left(\frac{1}{3}\right)^{k-1}\)

In Problems 13-16, expand \(f(z)=\frac{1}{(z-1)(z-2)}\) in a Laurent series valid for the indicated annular domain. $$ 0<|z-2|<1 $$

In Problems 13-24, determine the order of the poles for the given function. $$ f(z)=\tan z $$

Determine whether the given geometric series is convergent or divergent. If convergent, find its sum. \(\sum_{k=1}^{\infty}\left(\frac{i}{2}\right)^{k}\)

Expand the given function in a Taylor series centered at the indicated point. Give the radius of convergence of each series. . \(f(z)=\frac{z-1}{3-z}, z_{0}=1\)