II B.Tech I Semester Regular Examinations, November 2008

May 19th, 2015
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1. (a) Show that yq − 1 (log y )p −1 dy = Γ(p) q p where p>0, q>0. (b) Prove that β(m , 1 ) = 22m − 1 β(m , m ) 2. (a) Prove that the function f(z) = u + i v , where f (z) = x (1+i) + y 2 (1−i) , z = 0 and f(0) = 0 is continuous and that Cauchy’s Riemann equations are satisfied

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Code No: 07A3BS02Set No. 1II B.Tech I Semester Regular Examinations, November 2008MATHEMATICS-III( Common to Electrical & Electronic Engineering, Electronics &Communication Engineering, Electronics & Instrumentation Engineering,Electronics & Control Engineering, Electronics & Telematics, Electronics &Computer Engineering and Instrumentation & Control Engineering)Time: 3 hoursAnswer any FIVE QuestionsAll Questions carry equal marksMax Marks: 801. (a) Show thatyq 1 (log y )p 1 dy =(p)q pwhere p>0, q>0.(b) Prove that (m , 1 ) = 22m 1 (m , m )[8+8]3 32. (a) Prove that the function f(z) = u + i v , where f (z) = x (1+i) + y 2 (1i) , z =0 and f(0) = 0 is continuous and that Cauchys Riemann equations are satisedat the origin, yet f(z) does not exist.(b) Find the analytic function whose real part is y +ex cos y.3. (a) Prove thati. ii = e(4n+1) 2ii. log ii = (2n + 1 )(b) If tan( A + iB ) = x + iy , prove that x2 + y2 + 2x cot2A = 1.4. (a) Evaluatez 2dz, along0i. the line y = x/3ii. the parabola x =3y2(b) Use Cauchys integral formula to evaluatez 32z+1(zi)where c is the circle|z| = 2.5. (a) Expand f (z) =in a series of positive and negative powers of z.[10+6](b) Expand ez as Taylors series about z = 1.[8+8]6. Evaluate20d(53 sin )2using residue theorem.[16]7. Use Rouches theorem to show that the e

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