91Ó°ÊÓ

Expand \(f(z)=\frac{z}{(z+1)(z-2)}\) in a Laurent series valid for the indicated annular domain. \(|z+1|>3\)

In Problems 11-30, evaluate the Cauchy principal value of the given improper integral. $$ \int_{-\infty}^{\infty} \frac{d x}{\left(x^{2}+1\right)^{2}\left(x^{2}+9\right)} $$

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

In Problems 13-22, expand the given function in a Taylor series centered at the indicated point. Give the radius of convergence of each series. $$ f(z)=\frac{1+z}{1-z}, z_{0}=i $$

In Problems 15-20, determine whether the given geometric series is convergent or divergent. If convergent, find its sum. $$ \sum_{k=0}^{\infty} \frac{1}{2} i^{k} $$

In Problems 13-24, determine the order of the poles for the given function. $$ f(z)=\frac{\cot \pi z}{z^{2}} $$

Evaluate the Cauchy principal value of the given improper integral. \(\int_{-\infty}^{\infty} \frac{d x}{\left(x^{2}+1\right)^{2}\left(x^{2}+9\right)}\)

Determine whether the given geometric series is convergent or divergent. If convergent, find its sum. \(\sum_{k=0}^{\infty} \frac{1}{2} i^{k}\)

In Problems 17-20, use Cauchy's residue theorem, where appropriate, to evaluate the given integral along the indicated contours. \(\oint_{C} \frac{z+1}{z^{2}(z-2 i)} d z\) (a) \(|z|=1\) (b) \(|z-2 i|=1\) (c) \(|z-2 i|=4\)

Use Cauchy's residue theorem, where appropriate, to evaluate the given integral along the indicated contours. \(\oint_{C} \frac{z+1}{z^{2}(z-2 i)} d z\) (a) \(|z|=1\) (b) \(|z-2 i|=1\) (c) \(|z-2 i|=4\)