Applied Mathematics - 3 - Dec 2016

Electronics & Telecomm. (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) Determine the constants a, b, c, d, e if $ f(z)=\left ( ax^4+bx^2 y^2 +cy^4+dx^2-2y^2\right )+\left ( 4x^3 y-exy^3+4xy \right ) $/ is analytic.(5 marks) 1(b) Find half range Fourier sine series for f(x)=x², 0<x<3.< a="">

</x<3.<></span>(5 marks) 1(c) Find the directional derivative of $ \varphi \left ( x,y,z \right )=xy^2+yz^3 $/ at the point (2,-1,1) in the direction of the vector i + 2j + 2k.(5 marks) 1(d) Evaluate $$\int_{0}^{\infty }e^{-2t}t^{5}\cosh t\ dt.$$(5 marks) 2(a) Prove that $$\jmath_ \frac{3}{2}(x)=\sqrt{\frac{2}{\pi x}}\left ( \frac{\sin x}{x}-\cos x \right )$$(6 marks) 2(b) If f(z) = u + iv is analytic and $ u-v=e^x\left ( \cos y-\sin y \right )$/, fin f(z) in terms of z.(6 marks) 2(c) Obtain Fourier series for $\begin{align*} \begin{matrix} f(x)&= x+\frac{\pi }{2} &-\pi / Hence deduce that $\frac{\pi ^2}{8}=\frac{1}{1^2}+\frac{1}{3^2}+\frac{1}{s^2}+.....$\lt/span\gt\lt/span\gt\ltspan class='paper-ques-marks'\gt(8 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt3(a)\lt/b\gt Show that \ltspan class="math-tex"\gt$F=\left ( 2xy+z^3 \right )i+x^2j+3xz^2k $\lt/span\gt/, is a conservative field. Find its scalar potential and also find the work done by the force F in moving a praticle from (1,-2,1) to (3, 1, 4).\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt3(b)\lt/b\gt Show that the set of functions \ltspan class="math-tex"\gt$ \left { \sin \left ( 2n+1 \right )x \right },n=0, 1, 2,...$\lt/span\gt/ is orthogonal over [0,π/2}. Hence consturct orthonormal set of fucntions.\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt3(c)\lt/b\gt i)$$L^{-1}\left { \cot ^{-1}\left ( s+1 \right ) \right }$$
ii)$$L^{-1}\left ( \frac{e^{-2s}}{s^2+8s+25} \right )$$(8 marks) 4(a) Prove that $$\int \jmath _3(x)dx=\frac{2\jmath _1(x)}{x}-\jmath _2(x)$$(6 marks) 4(b) Find inverse Laplace of $ \frac{s}{\left ( s^2+a^2 \right )\left ( s^2+b^2 \right )}\left ( a\neq b \right ) $/ using Convolution theorem.(6 marks) 4(c) Expand f(x) = xsinx in the interval 0≤x≤2π as a Fourier series. Hence, deduce that $$\sum_{{n=2}}^{\infty }\ \ \frac{1}{n^2-1}=\frac{3}{4}$$(8 marks) 5(a) Using Gauss Diveragence theorem evaluate $\int \int _s\bar{N.}\bar{F}ds\ \ \text{where} \bar{F}=x^2i+zj+yzk $/ and S is the cube bounded by x=0, x=1, y=0, y=1, z=0, z-1(6 marks) 5(b) Prove that $$ \ j^{'}_{2}(x)=\left ( 1-\frac{4}{x^2} \right )\jmath _1(x)+\frac{2}{x}\jmath 0(x) \ ](6 marks) 5(c) Solve $ \left ( D^23D+2 \right )y=2\left ( t^2+t+1 \right )$/, with y(0)=2 and y(0)=0 by using Laplace transform(8 marks) 6(a) Evaluate by Green's theorem for $\int _c\left ( e^{-x}\sin dx+e^{-x} \cos y dy\right ) $/ where C is the rectangle whose vertices are (0,0), (π, 0), (π, π/2)(6 marks) 6(b) Show that under the transformation $ w=\frac{z-i}{z+i} $/ real axis in the z-plane is mapped onto the circle |w|=1(6 marks) 6(c) Find Fourier integral representation for $$f(x)\frac{e^{-ax}}{x}$$(8 marks)