All Questions page - 72

A subset \(S\) of a ring \(R\) is said to be multiplicative if \(ab \in S \;\forall \; a,b \in S\). Show that \(P\) is a prime ideal of a commutative ring \(R\) if and only if \(R-P\) is multiplicative.

Let \(I\) be the principal ideal \(\langle x \rangle \) of the ring \(\mathbb{Z}[x]\). Prove that \(I\) is prime but not maximal.

Let \(R\) be a commutative ring with identity in which the set of all non-units forms an ideal \(M\). Show that \(M\) is a maximal ideal of \(R\) and it contains every proper ideal of \(R\). Also show that for each \(r \in R\) either \(r\) or \(1-r\) is a unit.

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 \(|z|<1\)?

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.