GUJARAT TECHNOLOGICAL UNIVERSITY
SEMESTER-IV(NEW) — EXAMINATION - SUMMER 2019
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Subject Code:2141905 Date:09/05/2019
Subject Name: Complex Variables and Numerical Methods
Time:02:30 PM TO 05:30 PM Total Marks: 70
Instructions:
- Attempt all questions.
- Make suitable assumptions wherever necessary.
- Figures to the right indicate full marks.
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Q.1 (a) State De’Movier’s. Find arg [i/(-2-2i)]. 03
(b) Define the operators A,V and E. Prove that EV= A. 04
(c) State Cauchy — Riemann Equations. Show that 07
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(1) f(z) = sin z is everywhere analytic
(i1) f(z) = xy + iy is nowhere analytic.
Q.2 (a) State the formula for sin-1z. Find sin-1(-i) 03
(b) Find analytic function f(z) =u +iv, if u=2x(1-y). 04
(c) Classify the singularities of the analytic function. 07
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In each of the following case, identify the singular point and its type with justification.
(i) z/(z4+1) (ii) sin(1/z) (iii) (1 —coshz)/z5
OR
(c) Use residues to evaluate the improper integral: 07
?08 x2 dx / (x2+ 1)(x2+4)
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Q.3 (a) Evaluate the integral ?C z¯ dz when C is the right-hand half z=2ei?(-p/2 < ? < p/2), of the circle |z| = 2 from — 2i to 2i. 03
(b) Show that the mapping by w = 1/z transforms circles and lines into circles and lines. 04
(c) Give two Laurent series expansions in powers of z for the function f(z) = 1/[z(1 —2z)] and specify the regions in which those expansions are valid. 07
OR
Q.3 (a) Find the bilinear transformation which transforms z1 = 8, z2 =1, z3 = 0 into w1 =0, w2=1, w3 = 8 03
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(b) Determine and sketch the region : 04
(i) 0 < arg z < p/4, (ii) | z-3 | > 4,
Which of them are domains?
(c) State the Cauchy’s Integral Formula and its extension. Hence evaluate integral ?C (7z2-2z+5)/(z+1)3 dz, where C is circle |z - i|=2 07
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Q.4 (a) Evaluate ?37 x2log x dx taking four sub-intervals by trapezoidal rule. 03
(b) Apply Bisection method to find a real root of the equation 2x4 — 5x + 1 =0 correct to 2 decimal places. 04
(c) Given f(1)=22, f(2)=30, f(4)=82, f(7)=106, f(12)=206, find f(8) using Lagrange’s interpolation formula. 07
OR
Q.4 (a) The velocity of a car (running on a straight road) at intervals of 2 minutes are given below. 03
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Time (in min.): 0 2 4 6 8 10 12
Velocity (in km/hr): 0 22 30 27 18 7 0
Apply Simpson’s 1/3rd rule to find the distance covered by the car.
(b) Newton Raphson method find a root of the equation xsinx +cosx = 0 correct to four decimal places (taking initial guess X0 = p). 04
(c) State Striling’s Interpolation formula. 07
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Interpolate by means of Gauss’ forward formula the population for the year 1936 given the following data:
Year: 1901 1911 1921 1931 1941 1951
Population (1000s): 12 15 20 27 39 52
Q.5 (a) Using secant method find a real root of the equation x4 — 5x + 1 = 0 up to three iterations. 03
(b) Use Runge-Kutta method to solve y’ = xy, y(0)=1, for x = 0.2, with h=0.1. 04
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(c) Using Gauss-Seidel method to solve the following system correct to 3 decimal places: 83x + 11y —4z =95, 7x + 52y + 13z =104, 3x + 8y + 29z =71. 07
OR
Q.5 (a) Solve xlog10x =1.2 by Regula Falsi method correct to two decimal places. 03
(b) Solve y’=1-y, y(0)=0 in [0, 0.3] by modified Euler’s method taking h=0.1. 04
(c) Find the largest eigenvalue and corresponding eigenvector using power method, for 07
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A = | 4 4 2 |
| 1 4 1 |, taking X0 = (1, 1, 1)T.
| 2 1 3 |
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