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.

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.