## Applied Mathematics - 3 - Dec 2013

### Electronics Engineering (Semester 3)

TOTAL MARKS: 80

TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.

(2) Attempt any **three** from the remaining questions.

(3) Assume data if required.

(4) Figures to the right indicate full marks.
**1 (a)** Prove that real and imaginary parts of an analytic function F(z)=u+iv are harmonic function.(5 marks)
**1 (b)** Find the fourier series for f(x)=|sin x| in (-Π,Π).(5 marks)
**1 (c) ** Find the Laplace Transform of ?$$\int_0^t\ {ue}^{-3u}\sin{4}udu$$(5 marks)
**1 (d)** $$ if \ \bar{F}=xye^{2z}\widehat{i}+xy^2 \ \cos z \widehat{j}+x^2 \cos xy\ \widehat{k},\ find \ \bar{F} \ and \ curl \ \bar{F} $$(5 marks)
**2 (a)** Using laplace transofrm solve-

(D^{2} + 3D + 2) y= e^{-2t}. Sin t where y(0) = 0 and y' (0) =0.(6 marks)
**2 (b)** Find the directional derivative of d=x^{2} y cosz at (1,2, Π/2)in the direction of t=2i + 3j + 2k. (6 marks)
**2 (c) ** Find the Fourier expansion of $$f(x)=\sqrt{1-cosx}\ in\left(0,2\pi{}\right).Hence\ prove\ \frac{1}{2}=\sum_{n=1}^{\infty{}}\frac{1}{4n^2-1}.$$(8 marks)
**3 (a)** $$ Prove \ the \ J_{3/2}(x)= \sqrt{\dfrac{2}{\pi x}} \left \{\dfrac {\sin x}{x}- \cos x \right \} $$(6 marks)
**3 (b)** Evaluate by Green's theorem $$\oint_c\left(x^2ydx+y^3dy\right)$$ where C is closed path formed by y=x,y=x^{2}(6 marks)
**3 (c) ** i) Find the Laplace Transform of $$\frac{\cos{bt-\cos{at}}}{t}$$

ii) Find the Laplace Transform of $$\frac{d}{dt}\left[\frac{sint}{t}\right].$$

(8 marks)
**4 (a)** Show that the set of functions {sinx, sin3x.....} OR {sin(2n+1)x:n=0,1,2,3....} is orthogonal over [0, π/2],Hence construct orthonormal set of functions.(6 marks)
**4 (b)** Find the imaginary part whose real part is u= x^{3} - 3xy^{2} + 3x^{2} + 1(6 marks)
**4 (c) ** Find Inverse Laplace Transform of?

$$i)log\left(\frac{s^2+4}{s^2+9}\right)$$

$$ii)\frac{s}{\left(s^2+4\right)\left(s^2+9\right)}$$(8 marks)
**5 (a)** Obtain half range sine series for f(x)=x^{2} in 0<x<3.< a="">

</x<3.<><>(6 marks)
**5 (b)** A Vector field F is given by $$\bar{F}=\left(x^2-yz\right)\hat{i}+\left(y^2-zx\right)\hat{j}+\left(z^2-xy\right)\hat{k}$$ is irroational and hence find scalar point function ϕ such that F = Δ ϕ(6 marks)
**5 (c) ** Show that the function V=e^{x} (xsiny+ycosy) satisfies Laplace equation and find its corresponding analytic function (8 marks)
**6 (a)** By using stoke's theorem ,evaluate

$$\oint_c\left[\left(x^2+y^2\right)\hat{i}+\left(x^2-y^2\right)\hat{j}\right]\cdot d\bar{r}$$

where c is the boundary of a region enclosed by circles x^{2} + y^{2} =4, x^{2 }+ y^{2} = 16.(6 marks)
**6 (b)** Show that under the transformation w= 5-4z/4z-2 the circle |z|=1 in the z plane is transformed into a circle of unity in w-plane.(6 marks)
**6 (c) ** Prove that $$\int J_3\left(x\right)dx=\ -\frac{2J_1(x)}{x}-J_2(x)$$(8 marks)