1 (a)
Prove that f(z)= x

^{2}-y^{2}+2ixy is analytic and find f'(z).
5 M

1 (b)
Find the Fourier series expansion for f(x)=|x|, in (-?, ?).

5 M

1 (c)
Using Laplace transform solve the following differential equation with given condition \[ \dfrac {d^2y}{dt^2} + y=t, \] given that y(0)=1 & y'(0)=0.

5 M

1 (d)
\[ if \ \bar{A} =\nabla (xy + yz + zx), \ find \ \nabla \cdot \bar{A} \ and \ \nabla \time \bar{A} \]

5 M

2 (a)
\[ if \ L [j_0 (t) ] = \dfrac {1}{\sqrt{s^2+1}}\] prove that \[ \int^\infty_0 e^{-6 t} t j_0 (4 t) dt=3/500.

6 M

2 (b)
Find the directional derivative of ?=x

^{4}+y^{4}+z^{4}at A(1, -2, 1) in the direction of AB where B is (2,6,-1). Also find the maximum directional derivative of ? at (1, -2, 1).
6 M

2 (c)
Find the Fourier series expansion for f(x)=4-x

^{2}, in (0,2). Hence deduce that \[ \dfrac {\pi^2}{6} = \dfrac {1}{1^2} + \dfrac {1}{2^2} + \dfrac {1}{3^2} + \cdots \ \cdots \]
8 M

3 (a)
Prove that \[ J_{1/2} (x) = \sqrt{\dfrac {2}{\pi x}} \sin x \]

6 M

3 (b)
Using Green's theorem evaluate \[ \int_C (2x^2-y^2) dx + (x^2 + y^2) dy \] where 'C' is the boundary of the surface enclosed by the lines x=0, y=0, x=2, y=2.

6 M

3 (c) (i)
Find Laplace Transform of \[ e^{-3t} \int^1_0 u \sin 3u \ du\]

4 M

3 (c) (ii)
Find the Laplace transform of \[ \dfrac {d}{dt} \left ( \dfrac {1- \cos 2t}{t} \right ) \]

4 M

4 (a)
Obtain complex form of Fourier series for the functions f(x)=sin ax in (-?, ?), where a is not an integer.

6 M

4 (b)
Find the analytic function whose imaginary part is \[ v=\dfrac {x}{x^2+y^2} + cosh \ y\cdot \cos x \]

6 M

4 (c)
Find inverse Laplace Transform of following \[
i) \ \log \left ( \dfrac {s^2 + a^2}{\sqrt{s+b}} \right ) \\ ii) \ \dfrac {1} {s^3 (s-1)} \]

8 M

5 (a)
Obtain half-range cosine series for f(x)=x(2-x) in 0 < x< 2.

6 M

5 (b)
Prove that \[ \overline {F} = \dfrac {\overline r} {r^3} \] is both irrotational and solenoidal.

6 M

5 (c)
Show that the function u=sin x cosh y+2 cos x sinh y+x

^{2}-y^{2}+4xy satisfies Laplace's equation and find it corresponding analytic function.
8 M

6 (a)
Evaluate by Stroke's theorem \[ \int_c (x,y, \ dx + x \ y^2 \ dy ) \] where C is the square in the xy-plane with vertices (1,0), (0,1),(-1,0) and (0,-1).

6 M

6 (b)
Find the bilinear transformation, which maps the points z=-1, 1, &infity; onto the points w=-i, -1, i.

6 M

6 (c)
Show that the general solution of \[ \dfrac {d^2y}{d x^2} + 4x^2 y =0 \ is \ y =\sqrt{x} [A \ J_{1/4} (x^2) + B \ J_{-1/4} (x^2) ] \] where A and B are constants.

8 M

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