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  \(u^\prime = 3u^{2/3},u(0)=0 .\)