Chapter 8

Find \(\operatorname{Res} \mid f, 0]\) for (a) \(f(z)=z^{-1} \exp z\) (b) \(f(z)=z^{-3} \cosh 4 z\) (c) \(f(z)=\csc z\) (d) \(f(z)=\frac{z^{2}+4 z+5}{z^{2}+z}\) (e) \(f(z)=\cot z\) (f) \(f(z)=a^{-3} \cos z\) (g) \(f(z)=z^{-1} \sin z\) (h) \(f(z)=\frac{z^{2}+4 z+5}{z^{3}}\) (i) \(f(z)=\exp \left(1+\frac{1}{x}\right)\). (j) \(f(z)=z^{4} \sin \left(\frac{1}{z}\right)\). (k) \(f(z)=z^{-1} \csc z\) (1) \(f(z)=z^{-2} \csc z\) \((m) f(z)=\frac{\exp (4 z)-1}{\sin ^{2} z}\) (n) \(f(z)=z^{-1} \csc ^{2} z\).

Use residues to find $$ \int_{0}^{2 \pi} \frac{1}{3 \cos \theta+5} d \theta $$

Use residues to evaluate $$ \int_{-\infty}^{\infty} \frac{x^{2} d x}{\left(x^{2}+16\right)^{2}} $$

Use residues to find the Cauchy principal value of $$ \int_{-\infty}^{\infty} \frac{\cos x d x}{x^{2}+9} \text { and } \int_{-\infty}^{\infty} \frac{\sin x d x}{x^{2}+9} \text { . } $$

Use residues to compute $$ \text { P.V. } \int_{-\infty}^{\infty} \frac{d x}{x(x-1)(x-2)} \text { . } $$

Let \(f\) and \(g\) have an isolated singularity at \(z_{0}\). Show that \(\left.\operatorname{Res} \mid f+g, z_{0}\right]=\) \(\operatorname{Res}\left[f, z_{0}\right]+\operatorname{Res}\left[g, z_{0}\right]\)

Use residues to find $$ \int_{0}^{2 \pi} \frac{1}{4 \sin \theta+5} d \theta $$

Use residues to evaluate $$ \int_{-\infty}^{\infty} \frac{d x}{x^{2}+16} $$

Use residues to find the Cauchy principal value of $$ \int_{-\infty}^{\infty} \frac{x \cos x d x}{x^{2}+9} \text { and } \int_{-\infty}^{\infty} \frac{x \sin x d x}{x^{2}+9} . $$

Use residues to compute $$ \text { P.V. } \int_{-\infty}^{\infty} \frac{d x}{x^{3}+x} \text { . } $$

Show that four of the five roots of the equation \(z^{5}+15 z+1=0\) belong to the annulus \(A\left(\frac{3}{2}, 2,0\right)=\left\\{z: \frac{3}{2}<|z|<2\right\\}\).

Use residues to find $$ \int_{0}^{2 \pi} \frac{1}{15 \sin ^{2} \theta+1} d \theta $$

Use residues to evaluate $$ \int_{-\infty}^{\infty} \frac{x d x}{\left(x^{2}+9\right)^{2}} $$

Use residues to find the Cauchy principal value of $$ \int_{-\infty}^{\infty} \frac{x \sin x d x}{\left(x^{2}+4\right)^{2}} $$

Use residues to compute $$ \text { P.V. } \int_{-\infty}^{\infty} \frac{x d x}{x^{3}+1} \text { . } $$

Let \(g(z)=z^{5}+4 z-15\) (a) Show that there are no zeros in \(D_{1}(0)\). (b) Show that there are five zeros in \(D_{2}(0)\). Hint: Consider \(f(z)=-z^{5}\). Remark: A factorization of the polynomial using numerical approximations for the coefficients is \((z-1.546)\left(z^{2}-1.340 z+2.857\right)\left(z^{2}+2.885 z+3.397\right)\)

Let \(f\) and \(g\) be analytic at \(z_{0}\). If \(f\left(z_{0}\right) \neq 0\) and \(g\) has a simple zero at \(z_{0}\). then show that \(\operatorname{Res}\left[\frac{L}{g}, z_{0}\right]=\frac{f\left(z_{0}\right)}{g^{\prime}\left(z_{0}\right)}\).

Use residues to evaluate $$ \int_{-\infty}^{\infty} \frac{x+3}{\left(x^{2}+9\right)^{2}} d x $$

Use residues to find the Cauchy principal value of $$ \int_{-\infty}^{\infty} \frac{\cos x d x}{\left(x^{2}+4\right)^{2}} $$

Use residues to compute $$ \text { P.V. } \int_{-\infty}^{\infty} \frac{d x}{x^{3}+1} $$

Let \(g(z)=z^{3}+9 z+27\). (a) Show that there are no zeros in \(D_{2}(0)\). (b) Show that there are three zeros in \(D_{4}(0)\). Remark: A factorization of the polynomial using numerical approximations for the coefficients is \((z+2.047)\left(z^{2}-2.047 z+13.19\right)\)

Find \(\int_{C}(z-1)^{-2}\left(z^{2}+4\right)^{-1} d z\) when (a) \(C=C_{1}^{+}(1)\). (b) \(C=C_{4}^{+}(0)\).

Use residues to find $$ \int_{0}^{2 \pi} \frac{\sin ^{2} \theta}{5+4 \cos \theta} d \theta $$

Use residues to find the Cauchy principal value of $$ \int_{-\infty}^{\infty} \frac{\cos x d x}{\left(x^{2}+4\right)\left(x^{2}+9\right)} $$

Use residues to compute $$ \text { P.V. } \int_{-\infty}^{\infty} \frac{x^{2} d x}{x^{4}-1} \text { . } $$

Use residues to compute $$ \text { P.V. } \int_{0}^{\infty} \frac{\ln \left(x^{2}+1\right) d x}{x^{2}+1} \text { , Hint: Use the integrand } f(z)=\frac{\log (z+i)}{x^{2}+1} \text { , } $$

Let \(g(z)=z^{5}+6 z^{2}+2 z+1\) (a) Show that there are two zeros in \(D_{1}(0)\). (b) Show that there are five zeros in \(D_{2}(0)\).

Find \(\int_{C}\left(z^{6}+1\right)^{-1} d z\) when (a) \(C=C_{-}^{+}(i)\). (b) \(C=C_{1}^{+}\left(\frac{1+i}{2}\right) .\) Hint: If \(z_{0}\) is a singularity of \(f(z)=\frac{1}{z^{6}+1}\), show that \(\operatorname{Res}\left[f, z_{0}\right]=-\frac{1}{6} z_{0}\).

Use residues to find $$ \int_{0}^{2 \pi} \frac{\sin ^{2} \theta}{5-3 \cos \theta} d \theta $$

Use residues to find the Cauchy principal value of $$ \int_{-\infty}^{\infty} \frac{\cos x d x}{\left(x^{2}+1\right)\left(x^{2}+4\right)} $$

Use residues to compute $$ \text { P.V. } \int_{-\infty}^{\infty} \frac{x^{4} d x}{x^{6}-1} \text { . } $$

Let \(g(z)=z^{6}-5 z^{4}+10\). (a) Show that there are no zeros in \(|z|<1\). (b) Show that there are four zeros in \(|z|<2\). (c) Show that there are six zeros in \(|z|<3\).

Find \(\int_{C}\left(z^{6}+1\right)^{-1} d z\) when (a) \(C=C_{-}^{+}(i)\). (b) \(C=C_{1}^{+}\left(\frac{1+i}{2}\right) .\) Hint: If \(z_{0}\) is a singularity of \(f(z)=\frac{1}{z^{6}+1}\), show that \(\operatorname{Res}\left[f, z_{0}\right]=-\frac{1}{6} z_{0}\).

Use residues to find $$ \int_{0}^{2 \pi} \frac{1}{(5+3 \cos \theta)^{2}} d \theta $$

Use residues to find the Cauchy principal value of $$ \int_{-\infty}^{\infty} \frac{\cos x d x}{x^{2}-2 x+5} $$

Use residues to compute $$ \text { P.V. } \int_{-\infty}^{\infty} \frac{\sin x d x}{x} \text { . } $$

Let \(g(z)=3 z^{3}-2 i z^{2}+i z-7 .\) (a) Show that there are no zeros in \(|z|<1\). (b) Show that there are three zeros in \(|z|<2\).

Use residues to evaluate $$ \int_{-\infty}^{\infty} \frac{x^{2} d x}{\left(x^{2}+4\right)^{3}} $$

Use residues to find the Cauchy principal value of $$ \int_{-\infty}^{\infty} \frac{\cos x d x}{x^{2}-4 x+5} $$

Use residues to find the partial fraction representations of (a) \(\frac{1}{z^{2}+3 z+2}\) (b) \(\frac{3 z-3}{z^{2}-z-2}\) (c) \(\frac{z^{2}-7 z+4}{z^{2}(z+4)}\) (d) \(\frac{10 z}{\left(z^{2}+4\right)\left(z^{2}+9\right)}\) (e) \(\frac{2 x^{2}-3 z-1}{(z-1)^{3}}\) (f) \(\frac{z^{3}+3 z^{2}-z+1}{z(z+1)^{2}\left(z^{2}+1\right)}\)

Use residues to find $$ \int_{0}^{2 \pi} \frac{\cos 2 \theta}{5+3 \cos \theta} d \theta . $$

Use residues to evaluate $$ \int_{-\infty}^{\infty} \frac{d x}{\left(x^{2}+1\right)^{2}\left(x^{2}+4\right)} $$

Use residues to find the Cauchy principal value of $$ \int_{-\infty}^{\infty} \frac{x \sin x d x}{x^{4}+4} $$

Use residues to compute $$ \text { P.V. } \int_{-\infty}^{\infty} \frac{\sin x d x}{x\left(\pi^{2}-x^{2}\right)^{-}} $$

Suppose that \(h(z)\) is analytic and nonzero and \(|h(z)|<1\) for \(z \in D_{1}(0)\). Prove that the function \(g(z)=h(z)-z^{n}\) has \(n\) zeros inside the unit circle \(C_{1}(0)\).

Use residues to evaluate $$ \int_{-\infty}^{\infty} \frac{x+2}{\left(x^{2}+4\right)\left(x^{2}+9\right)} d x $$

Use residues to compute $$ \text { P.V. } \int_{-\infty}^{\infty} \frac{\cos x d x}{\pi^{2}-4 x^{2}} \text { . } $$

Suppose that \(f(z)\) is analytic inside and on the simple closed contour \(C\). If \(f(z)\) is a one-to-one function at points \(z\) on \(C\), then prove that \(f(z)\) is one-to-one inside C. Hint: Consider the image of \(C\).

Let \(f\) be analytic at the points \(0, \pm 1, \pm 2, \ldots\). If \(g(z)=\pi f(z) \cot \pi z\), then show that \(\operatorname{Res}[g, n]=f(n)\) for \(n=0, \pm 1, \pm 2, \ldots .\)

Use residues to find $$ \int_{0}^{2 \pi} \frac{1}{\left(1+3 \cos ^{2} \theta\right)^{2}} d \theta $$