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  $u^\prime = 3u^{2/3},u(0)=0 .$

(a) Show that all the solutions of the linear equation $u^\prime +a(t)u = 0, \ t \in [0,\infty)$, where $a:[0,\infty) \rightarrow [0,\infty)$ is a continuous function, are always bounded if $\int_0^\infty a(s)ds < \infty$ .

(b) Consider the equation $(1+t)^2u^\prime - u = 0, \ t \in [0,\infty)$ .

(i) Find the unique solution.

(ii) Show that $\mid u(0) \mid \le \mid u(t) \mid < \mid u(0) \mid e$, for all $t \in [0 ,\infty)$ . How you are connecting this result with the problem $(2) (a)$ ?

Let $u$ denotes the solution of the initial value problem:

                 $u^\prime +cu = d(t) , \ u(0)=0$ ,

where $c$ is a complex constant. Let $d$ be a continuous function and bounded by $M$ on

$[0,\infty)$. Then, prove that

                      $\mid u(t) \mid \le \frac{M}{\mid Re(c) \mid }, \ for \ all \ t \in [0,\infty).$

Let $c,d:I \rightarrow \mathbb{R}$ be continuous functions. Let $u_1,u_2$ denote the solutions of the initial value problem

                      $u^\prime + c(t)u = d(t)$ ,

with $u_1(t_0)=u_{10}$ and $u_2(t_0)=u_{20}$   on an interval $I$ containing $t_0$. Then, show that

$\mid u_1(t) - u_2(t) \mid= \mid u_{10} -u_{20} \mid \exp \left [ - \int_{t_0}^t c(s)ds \right ] ,t \in I ,$,

and, hence, show that $u_1(t) = u_1(t;t_0 , u_{10} )$ is a continuous function of $x_{10}$ for any fixed $t$ and $t_0$. Thus, conclude that $u(t) = u(t;t_0,u_0)$ of a linear equation is a continuous function of an initial parameter $u_0$.