Chapter 6

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

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

In Problems 1-12, 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, write out the first five terms of the given sequence. $$ \left\\{5 i^{n}\right\\} $$

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

Show that \(z=0\) is a removable singularity of the given function. Supply a definition of \(f(0)\) so that \(f\) is analytic at \(z=0\). $$ f(z)=\frac{z^{3}-4 z^{2}}{1-e^{z^{2} / 2}} $$

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. $$ f(z)=\frac{1}{z^{3}(1-z)^{3}} ; \operatorname{Res}(f(z), 0) $$

Use known results to expand the given function in a Maclaurin series. Give the radius of convergence \(R\) of each series. $$ f(z)=\frac{1}{4-2 z} $$

Write out the first five terms of the given sequence. $$ \left\\{2+(-i)^{n}\right\\} $$

Evaluate the given trigonometric integral. $$ \int_{0}^{2 \pi} \frac{\cos \theta}{3+\sin \theta} d \theta $$

Show that \(z=0\) is a removable singularity of the given function. Supply a definition of \(f(0)\) so that \(f\) is analytic at \(z=0\). $$ f(z)=\frac{\sin 4 z-4 z}{z^{2}} $$

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

Use an appropriate Laurent series to find the indicated residue. $$ f(z)=\frac{4 z-6}{z(2-z)} ; \operatorname{Res}(f(z), 0) $$

Expand the given function in a Laurent series valid for the given annular domain. $$ f(z)=e^{-1 / z^{2}}, 0<|z| $$

Use known results to expand the given function in a Maclaurin series. Give the radius of convergence \(R\) of each series. $$ f(z)=\frac{1}{(1+2 z)^{2}} $$

Write out the first five terms of the given sequence. $$ \left\\{1+e^{n \pi i}\right\\} $$

Evaluate the given trigonometric integral. $$ \int_{0}^{2 \pi} \frac{1}{1+3 \cos ^{2} \theta} d \theta $$

Show that \(z=0\) is a removable singularity of the given function. Supply a definition of \(f(0)\) so that \(f\) is analytic at \(z=0\). $$ f(z)=\frac{1-\frac{1}{2} z^{10}-\cos z^{5}}{\sin z^{2}} $$

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

Use an appropriate Laurent series to find the indicated residue. $$ f(z)=(z+3)^{2} \sin \left(\frac{2}{z+3}\right) ; \operatorname{Res}(f(z),-3) $$

Use known results to expand the given function in a Maclaurin series. Give the radius of convergence \(R\) of each series. $$ f(z)=\frac{z}{(1-z)^{3}} $$

Write out the first five terms of the given sequence. $$ \left\\{(1+i)^{n}\right\\}[\text { Hint: Write in polar form.] } $$

Evaluate the given trigonometric integral. $$ \int_{0}^{\pi} \frac{1}{2-\cos \theta} d \theta[\text { Hint }: \text { Let } t=2 \pi-\theta .] $$

In Problems 5-10, determine the zeros and their order for the given function. $$ f(z)=(z+2-i)^{2} $$

Use an appropriate Laurent series to find the indicated residue. $$ f(z)=e^{-2 / z^{2}} ; \operatorname{Res}(f(z), 0) $$

Use known results to expand the given function in a Maclaurin series. Give the radius of convergence \(R\) of each series. $$ f(z)=e^{-2 z} $$

In Problems 5-10, determine whether the given sequence converges or diverges. $$ \left\\{\frac{3 n i+2}{n+n i}\right\\} $$

Evaluate the given trigonometric integral. $$ \int_{0}^{\pi} \frac{1}{1+\sin ^{2} \theta} d \theta $$

Determine the zeros and their order for the given function. $$ f(z)=z^{4}-16 $$

Use an appropriate Laurent series to find the indicated residue. $$ f(z)=\frac{e^{-z}}{(z-2)^{2}} ; \operatorname{Res}(f(z), 2) $$

Use known results to expand the given function in a Maclaurin series. Give the radius of convergence \(R\) of each series. $$ f(z)=z e^{-z^{2}} $$

Determine whether the given sequence converges or diverges. $$ \left\\{\frac{n i+2^{n}}{3 n i+5^{n}}\right\\} $$

Evaluate the given trigonometric integral. $$ \int_{0}^{2 \pi} \frac{\sin ^{2} \theta}{5+4 \cos \theta} d \theta $$

Determine the zeros and their order for the given function. $$ f(z)=z^{4}+z^{2} $$

The Laplace transform is a linear transformation; that is, for constants \(\alpha\) and \(\beta\), $$ \mathscr{L}\\{\alpha f(t)+\beta g(t)\\}=\alpha \mathscr{L}\\{f(t)\\}+\beta \mathscr{L}\\{g(t)\\} $$ whenever both transforms exist. Use the linearity defined above along with the definitions $$ \sinh k t=\frac{e^{k t}-e^{-k t}}{2}, \quad \cosh k t=\frac{e^{k t}-e^{-k t}}{2} $$ \(k\) a real constant, to find \(\mathscr{L}\\{\sinh k t\\}\) and \(\mathscr{L}\\{\cosh k t\\}\).

In Problems 7-12, expand \(f(z)=\frac{1}{z(z-3)}\) in a Laurent series valid for the indicated annular domain. $$ 0<|z|<3 $$

Use known results to expand the given function in a Maclaurin series. Give the radius of convergence \(R\) of each series. $$ f(z)=\sinh z $$

Evaluate the given trigonometric integral. $$ \int_{0}^{2 \pi} \frac{\cos ^{2} \theta}{3-\sin \theta} d \theta $$

Determine the zeros and their order for the given function. $$ f(z)=\sin ^{2} z $$

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

Use known results to expand the given function in a Maclaurin series. Give the radius of convergence \(R\) of each series. $$ f(z)=\cosh z $$

Determine whether the given sequence converges or diverges. $$ \left\\{\frac{n\left(1+i^{n}\right)}{n+1}\right\\} $$

Evaluate the given trigonometric integral. $$ \int_{0}^{2 \pi} \frac{\cos 2 \theta}{5-4 \cos \theta} d \theta $$

Determine the zeros and their order for the given function. $$ f(z)=e^{2 z}-e^{z} $$

In Problems 9-18, use the theory of residues to compute the inverse Laplace transform \(\mathscr{L}^{-1}\\{F(s)\\}\) for the given function \(F(s)\). $$ \frac{1}{s^{6}} $$

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