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 ${\int }_0^{\pi }{\displaystyle \frac{\cos 2\theta }{1-2a\cos \theta +a^2}}d\theta$, $-1<a<1$ is given by:

a) ${\displaystyle \frac{2\pi a^2}{1-a^2}}$
b) ${\displaystyle \frac{\pi a^2}{1-2a^2}}$
c) ${\displaystyle \frac{a^2}{1-a^2}}$
d) ${\displaystyle \frac{\pi a^2}{1-a^2}}$

2) If $f(z)=u+iv$ 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
${\int }_0^{2\pi }{\displaystyle \frac{{\cos }^{2}3\theta }{1-2p\cos 2\theta +p^2}}d\theta ,0<p<1$

a) $\pi {\displaystyle \frac{1-p+p^2}{1-p^2}}$
b) $\pi {\displaystyle \frac{1-p+p^2}{1-2p}}$
c) $\pi {\displaystyle \frac{1-p+2p^2}{1-p}}$
d) $\pi {\displaystyle \frac{1-p+p^2}{1-p}}$

4) Using the method of contour integration, find the value of ${\int }_0^{\pi }{\displaystyle \frac{ad\theta }{a^2+{\sin }^2\theta }}$, $(a>0)$:

a) ${\displaystyle \frac{\pi }{\sqrt{1+2a^2}}}$
b) ${\displaystyle \frac{2\pi }{\sqrt{1+a^2}}}$
c) ${\displaystyle \frac{\pi }{\sqrt{1+a^2}}}$
d) ${\displaystyle \frac{1}{\sqrt{1+a^2}}}$

5) Show that when $0<|z-1|<2,$ the function $f(z)={\displaystyle \frac{z}{(z-1)(z-3)}}$ has the Laurent series expansion in powers of $(z-1)$ as:

a) ${\displaystyle \frac{-1}{2(z-1)}}-\sum _{n=0}^{\mathrm{\infty }}{\displaystyle \frac{(z-1{)}^n}{2^{n+2}}}$
b) ${\displaystyle \frac{-1}{2(z-1)}}-3\sum _{n=0}^{\mathrm{\infty }}{\displaystyle \frac{(z-1{)}^n}{2^{n+2}}}$
c) ${\displaystyle \frac{-1}{2(z-3)}}-\sum _{n=0}^{\mathrm{\infty }}{\displaystyle \frac{(z-3{)}^n}{2^{n+2}}}$
d) ${\displaystyle \frac{-1}{2(z-1)}}-3\sum _{n=0}^{\mathrm{\infty }}{\displaystyle \frac{(z-3{)}^n}{2^{n+2}}}$

6) The value of ${\int }_0^{\mathrm{\infty }}{\displaystyle \frac{\cos (ax)}{x^2+1}}dx$, $a>0$ is given by:

a) ${\displaystyle \frac{\pi }{2}}{e}^{-a}$
b) $\pi e^{-a}$
c) ${\displaystyle \frac{\pi }{2}}{e}^{-2a}$
d) ${\displaystyle \frac{\pi }{4}}{e}^{-a}$

7) The value of ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\displaystyle \frac{dx}{1+x^4}}={\displaystyle \frac{\pi }{\sqrt{2}}}$ is given by:

a) ${\displaystyle \frac{\pi }{2\sqrt{2}}}$
b) ${\displaystyle \frac{\pi }{\sqrt{2}}}$
c) ${\displaystyle \frac{\pi }{\sqrt{3}}}$
d) ${\displaystyle \frac{2\pi }{\sqrt{2}}}$

8) Find the value of ${\displaystyle \frac{1}{\pi }}{\int }_0^{\pi }{e}^{\cos \theta }\cos (\sin \theta -n\theta )d\theta$ using the Laurent’s series expansion of $e^{{\displaystyle \frac{1}{z}}}$.
a) ${\displaystyle \frac{1}{n!}}$
b) ${\displaystyle \frac{1}{(n-1)!}}$
c) ${\displaystyle \frac{1}{(n+1)!}}$
d) ${\displaystyle \frac{1}{(n+2)!}}$

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 $|z|\to \mathrm{\infty }$. If $f(2)=5$ and $f(-1)=2,$ find $f(z)$.

a) $f(z)={\displaystyle \frac{-4+3z+3z^2+z^3}{z^2(z-1)}}$
b) $f(z)={\displaystyle \frac{-4+2z+3z^2+z^3}{z^2(z-1)}}$
c) $f(z)={\displaystyle \frac{-4+z+3z^2+z^3}{z^2(z-1)}}$
d) $f(z)={\displaystyle \frac{-4+3z+4z^2+z^3}{z^2(z-1)}}$

10) Find the value of ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\displaystyle \frac{x\sin ax}{x^4+4}}dx$, $a>0$ using the Residue Theorem:

a) ${\displaystyle \frac{\pi }{2}}{e}^{-a}\sin a$
b) ${\displaystyle \frac{\pi }{2}}{e}^{-2a}\sin a$
c) ${\displaystyle \frac{\pi }{2}}{e}^{-a}\sin 2a$
d) ${\displaystyle \frac{\pi }{3}}{e}^{-a}\sin a$

11) The value of $\sum _{n=0}^{\mathrm{\infty }}{\left({\displaystyle \frac{x^n}{n!}}\right)}^2$ is given by:

a) ${\displaystyle \frac{1}{2\pi }}{\int }_0^{2\pi }{e}^{2xcos\theta }d\theta$
b) ${\displaystyle \frac{1}{2\pi }}{\int }_0^{2\pi }{e}^{2xsin\theta }d\theta$
c) ${\displaystyle \frac{1}{\pi }}{\int }_0^{2\pi }{e}^{2xcos\theta }d\theta$
d) ${\displaystyle \frac{1}{2\pi }}{\int }_0^{2\pi }{e}^{2xsin\theta }d\theta$

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 ${\int }_0^{\mathrm{\infty }}{\displaystyle \frac{{\log }_e(x^2+1)}{x^2+1}}dx$ is given by:

a) $2\pi \log 2$
b) $\pi \log 2$
c) $\log 2$
d) $0$

14) Let $C$ be a circle $|z|=2$ oriented counter-clockwise. Evaluate the integral ${\int }_C{\displaystyle \frac{\cosh \pi z}{z(z^2+1)}}dz$ 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 ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\displaystyle \frac{x^{4}dx}{1+x^8}}$ is given by:

a) ${\displaystyle \frac{\pi }{\sqrt{2}}}\sin {\displaystyle \frac{\pi }{84}}$
b) ${\displaystyle \frac{\pi }{\sqrt{2}}}\sin {\displaystyle \frac{\pi }{8}}$
c) ${\displaystyle \frac{\pi }{\sqrt{2}}}\sin {\displaystyle \frac{\pi }{2}}$
d) ${\displaystyle \frac{\pi }{\sqrt{2}}}\sin {\displaystyle \frac{\pi }{6}}$