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 \(\int_{0}^{\infty} \dfrac{x \sin m x}{a^{2}+x^{2}} d x=\dfrac{\pi}{2} e^{-m a}\).

[2017, 15M]

3) Let \(\gamma:[0,1] \rightarrow C\) be the curve \(\gamma(t)=e^{2 \pi i t}\), \(0 \leq t \leq 1\). Find, giving justifications, the value of the contour integral \(\int_{\gamma} \dfrac{d z}{4 z^{2}-1}\).

[2016, 15M]

4) Evaluate by contour integration, \(\int_{0}^{1} \dfrac{d x}{\left(x^{2}-x^{3}\right)^{1/3}}\).

[2012, 15M]

5) If \(\alpha\), \(\beta\), \(\gamma\) are real numbers such that \(\alpha^{2}>\beta^{2}+\gamma^{2}\) show that: \(\int_{0}^{2 \pi} \dfrac{d \theta}{\alpha+\beta \cos \theta+\gamma \sin \theta}\)=\(\dfrac{2 \pi}{\sqrt{\alpha^{2}-\beta^{2}-\gamma^{2}}}\).

[2009, 30M]

6.(i) Evaluate the line integral \(\oint f(z) d z\) 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 \(x=9\), \(-\pi \leq y \leq \pi\) of \(z-plane\) under exponential function.

[2010, 15M]

7) Evaluate \(\int_{C}\left[\dfrac{e^{2 z}}{z^{2}\left(z^{2}+2 z+2\right)}+\log (z-6)+\dfrac{1}{(z-4)^{2}}\right] d z\), where \(C\) is the circle \(\vert z \vert =3\). State the theorems you use in evaluating above integral.

[2008, 15M]

8) Using contour integration evaluate \(\int_{0}^{2 \pi} \dfrac{\cos ^{2} 3 \theta}{1-2 p \cos 2 \theta+p^{2}} d \theta\), \(0 < p < 1\).

[2004, 15M]

9) Use the method of contour integration to prove that \(\int_{0}^{\pi} \dfrac{a d \theta}{a^{2}+\sin ^{2} \theta}=\dfrac{\pi}{\sqrt{1+a^{2}}} (a>0)\).

[2003, 15M]

10) Establish, by contour integration, \(\int_{0}^{\infty} \dfrac{\cos (a x)}{x^{2}+1} d x=\dfrac{\pi}{2} e^{-a}\) where \(a \geq 0\).

[2002, 15M]

11) Show that \(\int_{-\infty}^{\infty} \dfrac{1}{1+x^{4}} d x=\dfrac{\pi}{\sqrt{2}}\).

[2001, 15M]