IFoS PYQs 1
2008
1) Evaluate \(\bar{z} \int_{c}=d z\) from \(z=0\) to \(z=4+z i\) along the curve given by $z=t^{2}+i t$$.
[10M]
2) Evaluate by contour integration \(\int_{0}^{2 n} \frac{d \theta}{1-2 a \sin \theta+a^{2}}\), \(0<a<1\).
[13M]
3) Find the residue of \(f(z)=\tan z\) at \(\pi / 2\).
[13M]
2007
1) If \(f(z)=u+i v\) is analytic and \(y=e^{-x}(x \sin y-y \cos y)\), then find \(v\) and \(f(z)\).
[10M]
2) Expand \(f(z)=\dfrac{1}{(z+1)(z+3)}\) In a Laurent series valid for - (i) \(1<\vert z \mid<3\) (ii) \(\vert z\vert >3\)
[10M]
3) Using residue theorem, evaluate \(\int_{0}^{2 \pi} \dfrac{d \theta}{(3-2 \cos \theta+\sin \theta)}\)
[10M]
2006
1) If \(f(z)\) is analytic, prove that \(\left(\dfrac{\partial^{2}}{\partial x^{2}}+\dfrac{\partial^{2}}{\partial y^{2}}\right)\vert f(z)\vert ^{2}=4\vert f^{\prime}(z)\vert ^{2}, z=x+i y\)
[10M]
2) Show that the transformation \(\omega=\dfrac{5-4 z}{4 z-2}\) maps the unit circle \(\vert z\vert =1\) into a circle of radius unity and centre at \(1 / 2\).
[10M]
3) Use contour integration technique to find the value of \(\int_{0}^{2 \pi} \dfrac{d \theta}{2+\cos \theta}\)
[10M]
2005
1) If \(f\) analytic, prove that \(\left(\dfrac{\partial^{2}}{\partial x^{2}}+\dfrac{\partial^{2}}{\partial y^{2}}\right)\vert f(z)\vert ^{2}=4\vert f^{\prime}(z)\vert ^{2}\)
[10M]
2) Show that the transformation \(w=\dfrac{5-4 z}{4 z-2}\) maps unit circle \(\vert z\vert =1\) onto a circle of radius unity and centre at \(-\dfrac{1}{2}\).
[10M]
3) Use contour integration technique to find the value of \(\int_{0}^{2 \pi} \dfrac{d \theta}{2+\cos \theta}\)
[10M]