Let \(R\) be a commutative ring. Show that \(a|b \Leftrightarrow Ra \supset Rb, \forall a,b \in R\). Further if \(R\) contains the identity, show that for any \(a \in R\), if \(Ra\) is maximal, then \(a\) is an irreducible element of \(R\).

Let \(R\) be a non-trivial ring with identity in which every subring is an ideal. Show that \(R \cong \mathbb{Z}\) or \(\mathbb{Z_n}\) for some \(n .\)

Write parametric equation of hypocycloid, epicycloid, epitrochoid, trochoid.

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 \(|z|<1\)?