Show that \(\int_{|z|=R}|(\sin z) / z||d z| \rightarrow \infty\) as $R \rightarrow \infty$.

Determine the residue at each singularity for the following functions. (a) \(\frac{1}{\cos z}\) (b) \(\frac{z}{(z-1)^{2}(z-2)}\) (c) \(z^{n} \cos \frac{1}{z}\)

Suppose \(f(z)\) is analytic at \(z_{0}\) with $f^{\prime}\left(z_{0}\right) \neq 0 .\( Show that there exists an analytic function \)g(z)\( such that \)f(g(z))=z$ in some neighborhood of \(z_{0}\). This is known as the inverse function theorem.

Find the principal part for the following Laurent series. (b) \(\frac{z^{2}}{z^{4}-1} \quad(0<|z+i|<\sqrt{2})\) (c) \(\frac{e^{z}}{z^{4}} \quad(|z|>0)\) (d) \(\frac{\sin z}{z^{4}} \quad(|z|>0)\) (e) \(\frac{1}{\tan ^{2} z}-\frac{1}{z^{2}} \quad(0<|z|<\pi / 2)\).

Use contour integration to evaluate (a) \(\int_{0}^{\infty} \frac{d x}{1+x^{6}}\) (b) \(\int_{0}^{\infty} \frac{x^{2}}{1+x^{4}} d x\) (c) \(\int_{-\infty}^{\infty} \frac{d x}{1+x+x^{2}}\) (d) \(\int_{-\infty}^{\infty} \frac{x}{\left(x^{2}+2 x+2\right)^{2}} d x\) (e) \(\int_{-\infty}^{\infty} \frac{\cos a x}{1+x+x^{2}} d x\) (f) \(\int_{0}^{\infty} \frac{x^{2}+1}{x^{4}+1} d x\) (g) $\int_{0}^{\infty} \frac{x^{2}}{\left(x^{2}+4\right)^{2}\left(x^{2}+9\right)} d x$ (h) $\int_{0}^{\infty} \frac{x^{2}}{\left(x^{2}+1\right)\left(x^{2}+4\right)} d x$.

Describe the singularity at \(z=\infty\) for the following functions. (a) \(\frac{2 z^{2}+1}{3 z^{2}-10}\) (b) \(\frac{z^{2}}{z+1}\) (c) \(\frac{z^{2}+10}{e^{z}}\) (d) \(\frac{e^{z}}{z^{2}+10}\) (e) \(\tan z-z\) (f) \(\frac{1}{z}+\sin z\).