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.

Prove that the function 

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

is logarithmically convex.

Evaluate the integral $\int_\gamma \frac{dz}{z^2 +1}$ where $\gamma(\theta) =2|\cos2\theta|e^{i\theta}$ for $0\le \theta \le 2\pi$.

Let $G = \mathbb{C} \backslash\{a,b\},a \neq b$ and $\gamma$ be the curve in the figure below Show that $n(\gamma;a) = n(\gamma ;b)$ 

Find all possible values of $\int_\gamma \frac{dz}{1+z^2}$ where $\gamma$ is any closed rectifiable curve in $\mathbb{C}$ not passing through $\pm i$.

Evaluate $\int_\gamma \frac{e^z -e^{-z}}{z^4}dz$ where $\gamma$ is one of the following curves.

Let $\alpha(t)=e^{it}$ and $\beta(t)=(3/2)+ 3e^{it}$ for $t$ in the interval $[0,2\pi]$. Show that $\alpha$ and $\beta$ are freely homotopic on $A=\mathbb{C}\backslash\{0\}$.

Note: Homotopy means deforming one path to another. Freely homotopic means defining a function involving both the paths, as given in the definition of homotopy.

Let $\alpha:[a,b] \mapsto A$ and $\beta :[a,b] \mapsto B$ are closed paths in a set $A$ that is starlike with respect to a point $z_0$. Show that $\alpha$ and $\beta$ are freely homotopic in $A$ .

Prove that

$u_c(t) = \left\{ \begin{array} 00 \ for \ 0 \le t \le c, \\ (t-3)^3, \ for \ c\le t < \infty , c>0, \end{array}\right.$

 is a solution of  $u^\prime = 3u^{2/3},u(0)=0 .$