Let $C$ be circle $|z| =R(R >1)$  oriented counterclockwise. Show that

$\left | \int_C \frac {Logz^2}{z^2}dz \right | \le 4\pi \left ( \frac{\pi + logR}{R} \right )$ and hence   $\lim_{R \to \infty} \int_C \frac{Logz^2}{z^2}dz =0$ .

Without evaluating the integral, show that $\left | \int_C \frac {1}{\overline{z}^2 + \overline{z} +1}dz \right| \le \frac{9\pi}{16}$  where $C$ is the arc of circle $|z|=3$ from $z=3$ to $z=3i$ lying in the first quadrant.

Find the zeros and poles of

$f(z) = \frac{1}{(z-a)(z-b)(e^{z-a} -1)}$

and hence

$\int_{-\infty}^{\infty} \frac{1}{(x-a)(x-b)(e^{x-a} -1)}dx$

where $a,b$ are real .

Let $f$ be the function defined by $f(z)=\frac{(z+5)(2z+1)}{z^2-4}$ . Find the zero and poles of $f,\frac{1}{f}, f^\prime$ and $\frac{f^\prime}{f}$ .

Show that the function $f$ defined by the series $f(z)=\sum ^{\infty}_{n=1}\frac{1}{(z+n)^2}$ is meromorphic on every bounded subset of $\mathbb{C}$, and find the residues at its poles.

Find the general location of the roots of $z^4+z^3+4z^2+2z+3$.

Let $w=f(z)$ be analytic in a neighbourhood of $D=\overline{B}(0;1)$ . If $|w|<1$ for $|z|=1$, prove that there is a unique $z$ with $|z|<1$ and $f(z)=z$.

Let $w=f(z)$ be analytic in a neighborhood of $D=\overline{B}(0;1)$ . If $|w| \le 1$ for $|z|=1$, find about the fixed points of $f(z)=z$ in $|z|<1$?

Let $f(z) = z+ \sum_{n=2}^\infty a_nz^n ,|z| <1$. If $\sum_{n=2}^\infty |a_n| \le 1$, then prove that $f$ is one-one,

(i) without using Rouche’s theorem and

(ii) using Rouche’s theorem.

Consider this famous question on real analysis. The function $f(x)=x\sin(\pi/x)$ for $x>0$ satisfying Rolle’s theorem for any interval $(1/n,1/(n+1)),n \in \mathbb{N}$. The discussion leads to finding the solutions $\pm x_1, \pm x_2 , \cdots$ of the equation $\tan(/pi/x)=\pi/x$ where $\frac{1}{k+1/2}<x<\frac{1}{k}$ . Show that this equation has no other solution in the complex plane. Hence find the sum of the series $\sum_{k=1}^\infty z_k^{-2}$ .

Compare the function 

$\Gamma(z)=\frac{e^{-\gamma z}}{z} \prod_{n=1}^\infty \left(\frac{n}{n+z} \right) e^{z/n}$

with the canonical theorem of entire functions and provide the inference.