91Ó°ÊÓ

Determine the order of the poles for the given function. \(f(z)=\tan z\)

In Problems 17-20, use Cauchy's residue theorem, where appropriate, to evaluate the given integral along the indicated contours. $$ \oint_{C} \frac{1}{(z-1)(z+2)^{2}} d z $$ (a) \(|z|=\frac{1}{2}\) (b) \(|z|=\frac{3}{2}\) (c) \(|z|=3\)

In Problems 11-30, evaluate the Cauchy principal value of the given improper integral. $$ \int_{-\infty}^{\infty} \frac{2 x^{2}-1}{x^{4}+5 x^{2}+4} d x $$

Use Cauchy's residue theorem, where appropriate, to evaluate the given integral along the indicated contours. \(\oint_{C} \frac{1}{(z-1)(z+2)^{2}} d z\) (a) \(|z|=\frac{1}{2}\) (b) \(|z|=\frac{3}{2}\) (c) \(|z|=3\)

Evaluate the Cauchy principal value of the given improper integral. \(\int_{-\infty}^{\infty} \frac{2 x^{2}-1}{x^{4}+5 x^{2}+4} d x\)

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

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. $$ f(z)=\frac{z-1}{3-z}, z_{0}=1 $$

Expand \(f(z)=\frac{z}{(z+1)(z-2)}\) in a Laurent series valid for the indicated annular domain. \(0<|z+1|<3\)

$$ \text { In Problems 17-20, expand } f(z)=\frac{z}{(z+1)(z-2)} \text { in a Laurent } $$ $$ 0<|z+1|<3 $$

Determine the order of the poles for the given function. \(f(z)=\frac{\cot \pi z}{z^{2}}\)