91Ó°ÊÓ

In Problems 1-6, expand the given function in a Laurent series valid for the given annular domain. $$ f(z)=\frac{\cos z}{z}, 0<|z| $$

In Problems 1-4, write out the first five terms of the given sequence. $$ \left\\{5 i^{n}\right\\} $$

In Problems 1-12, use known results to expand the given function in a Maclaurin series. Give the radius of convergence \(R\) of each series. $$ f(z)=\frac{z}{1+z} $$

In Problems 1-4, show that \(z=0\) is a removable singularity of the given function. Supply a definition of \(f(0)\) so that \(f\) is analytic at \(z=0\). $$ f(z)=\frac{e^{2 z}-1}{z} $$

In Problems 1-4, find the Laplace transform of the given function. Determine a condition on \(s\) that is sufficient to guarantee the existence of \(F(s)=\mathscr{L}\\{f(t)\\}\). $$ f(t)=e^{5 t} $$

In Problems 1-6, use an appropriate Laurent series to find the indicated residue. $$ f(z)=\frac{2}{(z-1)(z+4)} ; \operatorname{Res}(f(z), 1) $$

In Problems 1-12, evaluate the given trigonometric integral. $$ \int_{0}^{2 \pi} \frac{1}{1+0.5 \sin \theta} d \theta $$

Evaluate the given trigonometric integral. $$ \int_{0}^{2 \pi} \frac{1}{10-6 \cos \theta} d \theta $$

Find the Laplace transform of the given function. Determine a condition on \(s\) that is sufficient to guarantee the existence of \(F(s)=\mathscr{L}\\{f(t)\\}\). $$ f(t)=e^{(-2+3 i) t} $$

Use an appropriate Laurent series to find the indicated residue. $$ f(z)=\frac{1}{z^{3}(1-z)^{3}} ; \operatorname{Res}(f(z), 0) $$