91Ó°ÊÓ

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

Show that the given sequence \(\left\\{z_{n}\right\\}\) converges to a complex number \(L\) by computing \(\lim _{n \rightarrow \infty} \operatorname{Re}\left(z_{n}\right)\) and \(\lim _{n \rightarrow \infty} \operatorname{Im}\left(z_{n}\right)\). \(\left\\{\left(\frac{1+i}{4}\right)^{n}\right\\}\)

The indicated number is a zero of the given function. Use a Maclaurin or Taylor series to determine the order of the zero. \(f(z)=1-\pi i+z+e^{z} ; z=\pi i\)

In Problems 1-12, expand the given function in a Maclaurin series. Give the radius of convergence of each series. $$ f(z)=\cos ^{2} z $$

In Problems 11-30, evaluate the Cauchy principal value of the given improper integral. $$ \int_{-\infty}^{\infty} \frac{1}{x^{2}-6 x+25} d x $$

In Problems 11 and 12, show that the given sequence \(\left\\{z_{n}\right\\}\) converges to a complex number \(L\) by computing \(\lim _{n \rightarrow \infty} \operatorname{Re}\left(z_{n}\right)\) and \(\lim _{n \rightarrow \infty} \operatorname{Im}\left(z_{n}\right)\). $$ \left\\{\left(\frac{1+i}{4}\right)^{n}\right\\} $$

In Problems 7-12, expand \(f(z)=\frac{1}{z(z-3)}\) in a Laurent series valid for the indicated annular domain. $$ 1<|z+1|<4 $$

Expand the given function in a Maclaurin series. Give the radius of convergence of each series. \(f(z)=\cos ^{2} z\)

Use (1), (2), or (4) to find the residue at each pole of the given function. \(f(z)=\frac{2 z-1}{(z-1)^{4}(z+3)}\)

In Problems \(9-12\), the indicated number is a zero of the given function. Use a Maclaurin or Taylor series to determine the order of the zero. $$ f(z)=1-\pi i+z+e^{z} ; z=\pi i $$