91Ó°ÊÓ

Evaluate the Cauchy principal value of the given improper integral. \(\int_{-\infty}^{\infty} \frac{1}{\left(x^{2}+1\right)^{3}} d x\)

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

In Problems 13-24, determine the order of the poles for the given function. $$ f(z)=\frac{1+4 i}{(z+2)(z+i)^{4}} $$

Determine the order of the poles for the given function. \(f(z)=\frac{1+4 i}{(z+2)(z+i)^{4}}\)

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

Expand the given function in a Taylor series centered at the indicated point. Give the radius of convergence of each series. \(f(z)=\frac{1}{3-z}, z_{0}=2 i\)

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

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 11-30, evaluate the Cauchy principal value of the given improper integral. $$ \int_{-\infty}^{\infty} \frac{x}{\left(x^{2}+4\right)^{3}} d x $$

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