Applied Mathematics 4 - Dec 2012

Electronics & Telecomm. (Semester 4)

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) Find the analytic u + iv given [u+v=e^{x} (cos y + sin y)+dfrac{x-y}{x+y}$$\lt/span\gt\ltspan class='paper-ques-marks'\gt(5 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt1(b)\lt/b\gt The matrix A is given by [A=egin{bmatrix} 1 & 0 &-3 \ 0& 3 &2 \ 0& 0&-2 end{bmatrix}$$.Find the eigen values and eigen vectors of B where [B=I-6A^{-1}$$\lt/span\gt\ltspan class='paper-ques-marks'\gt(5 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt1(c)\lt/b\gt Evaluate [int_{c}ar{f}.ar{dr}$$ along the arc of the curve form the point (1,0)to (e^2π,0)
where [?ar{f}=dfrac{xi+yj}{(x^{2}+y^{2})^{3/2}}$$ and curve C is [ar{r}=e^{t}i+e^{t}sin t j$$(5 marks) 1(d) Prove that [int J_{3}(X)dx=-dfrac{2}{x}J_{1}(x)-J_{2}(x).$$\lt/span\gt\ltspan class='paper-ques-marks'\gt(5 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt2(a)\lt/b\gt Find thhe Bilinear transformation which maps 1,-1,&infty; onto 1+i.1-I,1.Find its fixed points.\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt2(b)\lt/b\gt Evaluate A\ltsup\gt100\lt/sup\gt for [A=egin{bmatrix} 1 & 0& 0\ 1&0 &1 \ 0&1 &0 end{bmatrix}$$.(6 marks) 2(c) Verify Green's theorem for [?ar{f}=(x^{2}-xy)i+(x^{2}-y^{2})j$$ and c in Δ\ltsup\gtle\lt/sup\gtwith vertices (0,0),(1,1)&(1,-1).\lt/span\gt\ltspan class='paper-ques-marks'\gt(8 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt3(a)\lt/b\gt Show that f(x)=x\ltsup\gt2\lt/sup\gt,o\ltx\lt2, [f(x)="sum_{i-1}^{infty}dfrac{2(lambda_{i}^{2}-4)}{lambda_{i}^{3}J_{1}(lambda_{i})}Jo(lambda_{i}x)$$where[" lambda_{i}$$,i="0,1,2?..are" roots="" of="" jo(λ)="0\lt/a"\gt\ltbr\gt\ltbr\gt \lt/x\lt2,\gt\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt3(b)\lt/b\gt Show that [dfrac{x}{x^{2}+y^{2}}+2tan^{-1}left(dfrac{y}{x} ight)$$is imaginary part of an analytic function,find its real part and hence find the analytic function.(6 marks) 3(c) Evaluate [int_{c} dfrac{z^{2}}{z^{4}-1}dz$$\ltbr\gtc is \ltbr\gt(i)|z-1|=dfrac{1}{2} \ltbr\gt(ii)|z-1|=1 \ltbr\gt(iii)|z+i|=1\lt/span\gt\ltspan class='paper-ques-marks'\gt(8 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt4(a)\lt/b\gt Evaluate using stokes theorem [int_{c}y dx +zdy+xdz$$,where c is the curve of intersection of surfaces [x^{2}+y^{2}+z^{2}=a^{2} and x+z=a$$\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt4(b)\lt/b\gt Evaluate [int_{0}^{infty} dfrac{1}{x^{4}+1}dx$$(6 marks) 4(c) Find an orthogonal transformation which reduces the quadratic form [2x^{2}+y^{2}-3z^{2}-8xy-4xz+12xy $$to a diagonal form.find the rank, index,signature and class value of the given form.\lt/span\gt\ltspan class='paper-ques-marks'\gt(8 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt5(a)\lt/b\gt Prove that [J_{3/2}{x} = sqrt{dfrac{2}{pi x}}left(dfrac{sin x}{x}-cos x ight).\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt5(b)\lt/b\gt Find a minimal polynomial of A hence find\ltbr\gt[A^{10} where A= egin{bmatrix} 5 &-6 &-6 \ -1&4 &2 \ 3& -6 &4 end{bmatrix} $$(6 marks) 5(c) Find all possible Laurents series expansion of [dfrac{4z^{2}+2z-4}{z^{3}-4z}$$about z=2 and specify their domain of convergence.\lt/span\gt\ltspan class='paper-ques-marks'\gt(8 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt6(a)\lt/b\gt Prove that [2J_{n}^{1}(X) =J_{n-1}^{x}-J_{n+1}^{x}$$(6 marks) 6(b) Evaluate [int_{0}^{2pi}dfrac{cos 3 heta }{5-4 cos heta}d heta$$\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt6(c)\lt/b\gt Verify Gauss divergence theorem for F =xi + yj+z\ltsup\gt2\lt/sup\gtk,s in the surface bounded by the x\ltsup\gt2\lt/sup\gt+y\ltsup\gt2\lt/sup\gt=z\ltsup\gt2\lt/sup\gt and plane z=1.\lt/span\gt\ltspan class='paper-ques-marks'\gt(8 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt7(a)\lt/b\gt Show that under the transmission w =z\ltsup\gt2\lt/sup\gt,the circle |z-1|=1 is mapped onto cardiode ρ=2(1+cosφ) where w=ρe\ltsup\gtiφ\lt/sup\gt in w plane.\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt7(b)\lt/b\gt Find the matrix represented by A\ltsup\gt8\lt/sup\gt-5A\ltsup\gt7\lt/sup\gt+7A\ltsup\gt6\lt/sup\gt-3A\ltsup\gt5\lt/sup\gt+A\ltsup\gt4\lt/sup\gt-5A\ltsup\gt3\lt/sup\gt+8A\ltsup\gt2\lt/sup\gt-2A+I \ltbr\gt where [A=egin{bmatrix} 2 & 1 &1 \ 0& 1 & 0\ 1& 1 & 2 end{bmatrix}$$(6 marks) 7(c)(i) State and prove the Cauchy residue theorem.(4 marks) 7(c)(ii) Evaluate [int_{c} z^{6} e^{frac{-1}{x^{2}}} dz; c:|z|=1$$(4 marks)