Test 3: Complex Analysis
Instruction: Select the correct option corresponding to questions given below in the form shared at the bottom
Total Marks: 75
1) The value of \(\int_{0}^{\pi} \dfrac{\cos 2 \theta}{1-2 a \cos \theta+a^{2}} d \theta\), \(-1<a<1\) is given by:
a) \(\dfrac{2 \pi a^{2}}{1-a^{2}}\)
b) \(\dfrac{\pi a^{2}}{1-2a^{2}}\)
c) \(\dfrac{a^{2}}{1-a^{2}}\)
d) \(\dfrac{\pi a^{2}}{1-a^{2}}\)
2) If \(f(z)=u+i v\) is an analytic function of the complex variable \(z\) and \(u-v=e^{x}(\cos y-\sin y),\) determine \(f(z)\) in terms of \(z\).
a) \(f(z)=e^{z}\)
b) \(f(z)=e^{z} \sin z\)
c) \(f(z)=e^{z} \cos z\)
d) \(f(z)=e^{\pi z}\)
3) Using Contour integration, evaluate
\(\int_{0}^{2 \pi} \dfrac{\cos^{2} 3 \theta}{1-2 p \cos 2 \theta+p^{2}} d \theta, \quad 0<p<1\)
a) \(\pi \dfrac{1-p+p^{2}}{1-p^2}\)
b) \(\pi \dfrac{1-p+p^{2}}{1-2p}\)
c) \(\pi \dfrac{1-p+2p^{2}}{1-p}\)
d) \(\pi \dfrac{1-p+p^{2}}{1-p}\)
4) Using the method of contour integration, find the value of \(\int_{0}^{\pi} \dfrac{a d \theta}{a^{2}+\sin ^{2} \theta}\), \((a>0)\):
a) \(\dfrac{\pi}{\sqrt{1+2a^{2}}}\)
b) \(\dfrac{2\pi}{\sqrt{1+a^{2}}}\)
c) \(\dfrac{\pi}{\sqrt{1+a^{2}}}\)
d) \(\dfrac{1}{\sqrt{1+a^{2}}}\)
5) Show that when \(0<\vert z-1 \vert<2,\) the function \(f(z)=\dfrac{z}{(z-1)(z-3)}\) has the Laurent series expansion in powers of \((z-1)\) as:
a) \(\dfrac{-1}{2(z-1)}- \sum_{n=0}^{\infty} \dfrac{(z-1)^{n}}{2^{n+2}}\)
b) \(\dfrac{-1}{2(z-1)}-3 \sum_{n=0}^{\infty} \dfrac{(z-1)^{n}}{2^{n+2}}\)
c) \(\dfrac{-1}{2(z-3)}- \sum_{n=0}^{\infty} \dfrac{(z-3)^{n}}{2^{n+2}}\)
d) \(\dfrac{-1}{2(z-1)}-3 \sum_{n=0}^{\infty} \dfrac{(z-3)^{n}}{2^{n+2}}\)
6) The value of \(\int_{0}^{\infty} \dfrac{\cos (a x)}{x^{2}+1} d x\), \(a >0\) is given by:
a) \(\dfrac{\pi}{2} e^{-a}\)
b) \(\pi e^{-a}\)
c) \(\dfrac{\pi}{2} e^{-2a}\)
d) \(\dfrac{\pi}{4} e^{-a}\)
7) The value of \(\int_{-\infty}^{\infty} \dfrac{d x}{1+x^{4}}=\dfrac{\pi}{\sqrt{2}}\) is given by:
a) \(\dfrac{\pi}{2 \sqrt{2}}\)
b) \(\dfrac{\pi}{\sqrt{2}}\)
c) \(\dfrac{\pi}{\sqrt{3}}\)
d) \(\dfrac{2 \pi}{\sqrt{2}}\)
8) Find the value of \(\dfrac{1}{\pi} \int_{0}^{\pi} e^{\cos \theta} \cos (\sin \theta-n \theta) d \theta\) using the Laurent’s series expansion of \(e^{\dfrac{1}{z}}\).
a) \(\dfrac{1}{n!}\)
b) \(\dfrac{1}{(n-1)!}\)
c) \(\dfrac{1}{(n+1)!}\)
d) \(\dfrac{1}{(n+2)!}\)
9) The function \(f(z)\) has a double pole at \(z=0\) with residue \(2,\) a simple pole at \(z=1\) with residue \(2,\) is analytic at all other finite points of the plane and is bounded as \(\vert z \vert \rightarrow \infty\). If \(f(2)=5\) and \(f(-1)=2,\) find \(f(z)\).
a) \(f(z)=\dfrac{-4+3 z+3 z^{2}+z^{3}}{z^{2}(z-1)}\)
b) \(f(z)=\dfrac{-4+2 z+3 z^{2}+z^{3}}{z^{2}(z-1)}\)
c) \(f(z)=\dfrac{-4+z+3 z^{2}+z^{3}}{z^{2}(z-1)}\)
d) \(f(z)=\dfrac{-4+3z+4 z^{2}+z^{3}}{z^{2}(z-1)}\)
10) Find the value of \(\int_{-\infty}^{\infty} \dfrac{x \sin a x}{x^{4}+4} d x\), \(a>0\) using the Residue Theorem:
a) \(\dfrac{\pi}{2} e^{-a} \sin a\)
b) \(\dfrac{\pi}{2} e^{-2a} \sin a\)
c) \(\dfrac{\pi}{2} e^{-a} \sin 2a\)
d) \(\dfrac{\pi}{3} e^{-a} \sin a\)
11) The value of \(\sum_{n=0}^{\infty}\left(\dfrac{x^{n}}{n !}\right)^{2}\) is given by:
a) \(\dfrac{1}{2 \pi} \int_{0}^{2 \pi} e^{2 x cos \theta} d \theta\)
b) \(\dfrac{1}{2 \pi} \int_{0}^{2 \pi} e^{2 x sin \theta} d \theta\)
c) \(\dfrac{1}{\pi} \int_{0}^{2 \pi} e^{2 x cos \theta} d \theta\)
d) \(\dfrac{1}{2 \pi} \int_{0}^{2 \pi} e^{2 x sin \theta} d \theta\)
12) The analytic function whose real part is \(e^{x}(x \cos y-y \sin y)\) is given by:
a) \(e^{z} \cos z\)
b) \(e^{z} \sin z\)
c) \(e^{z}\)
d) \(z e^{z}\)
13) The value of \(\int_{0}^{\infty} \dfrac{\log _{e}\left(x^{2}+1\right)}{x^{2}+1} d x\) is given by:
a) \(2 \pi \log 2\)
b) \(\pi \log 2\)
c) \(\log 2\)
d) \(0\)
14) Let \(C\) be a circle \(\vert z \vert=2\) oriented counter-clockwise. Evaluate the integral \(\int_{C} \dfrac{\cosh \pi z}{z\left(z^{2}+1\right)} d z\) with the aid of residues.
a) \(4 \pi i\)
b) \(2 \pi i\)
c) \(\pi i\)
d) \(\sqrt{2} \pi i\)
15) The value of \(\int_{-\infty}^{\infty} \dfrac{x^{4} d x}{1+x^{8}}\) is given by:
a) \(\dfrac{\pi}{\sqrt{2}} \sin \dfrac{\pi}{84}\)
b) \(\dfrac{\pi}{\sqrt{2}} \sin \dfrac{\pi}{8}\)
c) \(\dfrac{\pi}{\sqrt{2}} \sin \dfrac{\pi}{2}\)
d) \(\dfrac{\pi}{\sqrt{2}} \sin \dfrac{\pi}{6}\)