Chapter 19

Expand the given function in a Maclaurin series. Give the radius of convergence of each series. \(f(z)=\frac{z}{1+z}\)

Write out the first five terms of the given sequence. \(\left\\{5 i^{n}\right\\}\)

Evaluate the given trigonometric integral. \(\int_{0}^{2 \pi} \frac{1}{1+0.5 \sin \theta} d \theta\)

Use a Laurent series to find the indicated residue. \(f(z)=\frac{2}{(z-1)(z+4)} ; \operatorname{Res}(f(z), 1)\)

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 1-10, evaluate the given trigonometric integral. $$ \int_{0}^{2 \pi} \frac{1}{1+0.5 \sin \theta} d \theta $$

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

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

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

Expand the given function in a Maclaurin series. Give the radius of convergence 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. \(\int_{0}^{2 \pi} \frac{1}{10-6 \cos \theta} d \theta\)

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

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

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

In Problems 1-6, use a Laurent series to find the indicated residue. $$ f(z)=\frac{1}{z^{3}(1-z)^{3}} ; \operatorname{Res}(f(z), 0) $$

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

In Problems 1-12, expand the given function in a Maclaurin series. Give the radius of convergence of each series. $$ f(z)=\frac{1}{4-2 z} $$

In Problems 1-4, write out the first five terms of the given sequence. $$ \left\\{2+(-i)^{n}\right\\} $$

Expand the given function in a Maclaurin series. Give the radius of convergence 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. \(\int_{0}^{2 \pi} \frac{\cos \theta}{3+\sin \theta} d \theta\)

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

Determine the zeros and their orders for the given function. \(f(z)=(z+2-i)^{2}\)

In Problems 1-10, evaluate the given trigonometric integral. $$ \int_{0}^{2 \pi} \frac{\cos \theta}{3+\sin \theta} d \theta $$

In Problems 1-6, use a Laurent series to find the indicated residue. $$ f(z)=\frac{4 z-6}{z(2-z)} ; \operatorname{Res}(f(z), 0) $$

In Problems 3-8, determine the zeros and their orders for the given function. $$ f(z)=(z+2-i)^{2} $$

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

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

Expand the given function in a Maclaurin series. Give the radius of convergence of each series. \(f(z)=\frac{z}{(1-z)^{3}}\)

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

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

Use a Laurent series to find the indicated residue. \(f(z)=(z+3)^{2} \sin \frac{2}{z+3} ;\) Res \((f(z),-3)\)

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

In Problems 1-10, evaluate the given trigonometric integral. $$ \int_{0}^{2 \pi} \frac{1}{1+3 \cos ^{2} \theta} d \theta $$

In Problems 1-6, use a Laurent series to find the indicated residue. $$ f(z)=(z+3)^{2} \sin \frac{2}{z+3} ; \operatorname{Res}(f(z),-3) $$

In Problems 3-8, determine the zeros and their orders for the given function. $$ f(z)=z^{4}-16 $$

In Problems 1-12, expand the given function in a Maclaurin series. Give the radius of convergence of each series. $$ f(z)=\frac{z}{(1-z)^{3}} $$

In Problems 1-4, write out the first five terms of the given sequence. $$ \left\\{(1+i)^{n}\right\\} \text { [Hint: Write in polar form.] } $$

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

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

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

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

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

In Problems 1-10, evaluate the given trigonometric integral. $$ \int_{0}^{\pi} \frac{1}{2-\cos \theta} d \theta $$

In Problems 1-6, use a Laurent series to find the indicated residue. $$ f(z)=e^{-2 / z^{2}} ; \operatorname{Res}(f(z), 0) $$

In Problems 3-8, determine the zeros and their orders for the given function. $$ f(z)=z^{4}+z^{2} $$

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

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