PYQs

1) Evalute the integral ${\int }_{C}Re(z^2)dz$ from 0 to $2+4i$ along the curve $C$ where $C$ is a parabola $y=x^2$.

[2019, 10M]

2) Using contour integral method, prove that ${\int }_0^{\mathrm{\infty }}{\displaystyle \frac{x\sin mx}{a^2+x^2}}dx={\displaystyle \frac{\pi }{2}}{e}^{-ma}$.

[2017, 15M]

3) Let $\gamma :[0,1]\to C$ be the curve $\gamma (t)=e^{2\pi it}$, $0\le t\le 1$. Find, giving justifications, the value of the contour integral ${\int }_{\gamma }{\displaystyle \frac{dz}{4z^2-1}}$.

[2016, 15M]

4) Evaluate by contour integration, ${\int }_0^1{\displaystyle \frac{dx}{{(x^2-x^3)}^{1/3}}}$.

[2012, 15M]

5) If $\alpha$, $\beta$, $\gamma$ are real numbers such that ${\alpha }^2>{\beta }^2+{\gamma }^2$ show that: ${\int }_0^{2\pi }{\displaystyle \frac{d\theta }{\alpha +\beta \cos \theta +\gamma \sin \theta }}$=${\displaystyle \frac{2\pi }{\sqrt{{\alpha }^2-{\beta }^2-{\gamma }^2}}}$.

[2009, 30M]

6.(i) Evaluate the line integral $\oint f(z)dz$ where $f(z)=z^2$, $c$ is the boundary of the triangle with vertices $A(0,0)$, $B(1,0)$, $C(1,2)$ in that order.

6.(ii) Find the image of the finite vertical strip $R$: $x=5$ to $x=9$, $-\pi \le y\le \pi$ of $z-plane$ under exponential function.

[2010, 15M]

7) Evaluate ${\int }_C[{\displaystyle \frac{e^{2z}}{z^2(z^2+2z+2)}}+\log (z-6)+{\displaystyle \frac{1}{(z-4{)}^2}}]dz$, where $C$ is the circle $|z|=3$. State the theorems you use in evaluating above integral.

[2008, 15M]

8) Using contour integration evaluate ${\int }_0^{2\pi }{\displaystyle \frac{{\cos }^{2}3\theta }{1-2p\cos 2\theta +p^2}}d\theta$, $0<p<1$.

[2004, 15M]

9) Use the method of contour integration to prove that ${\int }_0^{\pi }{\displaystyle \frac{ad\theta }{a^2+{\sin }^2\theta }}={\displaystyle \frac{\pi }{\sqrt{1+a^2}}}(a>0)$.

[2003, 15M]

10) Establish, by contour integration, ${\int }_0^{\mathrm{\infty }}{\displaystyle \frac{\cos (ax)}{x^2+1}}dx={\displaystyle \frac{\pi }{2}}{e}^{-a}$ where $a\ge 0$.

[2002, 15M]

11) Show that ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\displaystyle \frac{1}{1+x^4}}dx={\displaystyle \frac{\pi }{\sqrt{2}}}$.

[2001, 15M]