PYQs 4

We will cover following topics

2008
2007
2006
2005

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 $\int_{0}^{2 n} \frac{d θ}{1 - 2 a \sin θ + a^{2}}$ , $0 < a < 1$ .

[13M]

3) Find the residue of $f (z) = \tan z$ at $π / 2$ .

[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 $f (z) = \frac{1}{(z + 1) (z + 3)}$ In a Laurent series valid for - (i) $1 < | z ∣< 3$ (ii) $| z | > 3$

[10M]

3) Using residue theorem, evaluate $\int_{0}^{2 π} \frac{d θ}{(3 - 2 \cos θ + \sin θ)}$

[10M]

2006

1) If $f (z)$ is analytic, prove that $(\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}) | f (z) |^{2} = 4 | f^{'} (z) |^{2}, z = x + i y$

[10M]

2) Show that the transformation $ω = \frac{5 - 4 z}{4 z - 2}$ maps the unit circle $| z | = 1$ into a circle of radius unity and centre at $1 / 2$ .

[10M]

3) Use contour integration technique to find the value of $\int_{0}^{2 π} \frac{d θ}{2 + \cos θ}$

[10M]

2005

1) If $f$ analytic, prove that $(\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}) | f (z) |^{2} = 4 | f^{'} (z) |^{2}$

[10M]

2) Show that the transformation $w = \frac{5 - 4 z}{4 z - 2}$ maps unit circle $| z | = 1$ onto a circle of radius unity and centre at $- \frac{1}{2}$ .

[10M]

3) Use contour integration technique to find the value of $\int_{0}^{2 π} \frac{d θ}{2 + \cos θ}$

[10M]

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