GUJARAT TECHNOLOGICAL UNIVERSITY
SEMESTER- IV EXAMINATION - SUMMER 2020
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Subject Code: 2141905 Date:28/10/2020
Subject Name: COMPLEX VARIABLES AND NUMERICAL METHODS
Time: 10:30 AM TO 01:30 PM Total Marks: 70
Instructions:
- Attempt all questions.
- Make suitable assumptions wherever necessary.
- Figures to the right indicate full marks.
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MARKS
Q.1 (a) Verify Cauchy-Riemann equation for f(z)=cosxcosh y—isinxsinhy . 03
(b) Find all cube roots of complex number (—8i ) . 04
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(c) Evaluate ? dx with n=1 by (i) Trapezoidal Rule (ii) Simpson’s ? Rule 07
(iii) Simpson’s ? Rule.
Q.2 (a) Find the principal value of (1—i )4i . 03
(b) Using Parametric representation of C, evaluate ?z dz; C is the circle 04
z=2eit(0<t<2p)
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(c) Show that u(x,y)=2x—x3+3xy2 is harmonic function and find harmonic 07
conjugate v(x,y).
OR
(c) For f(z)= z?0 07
0 ; z=0
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Show that C-R equation is satisfied at origin but f'(0) does not exist.
Q.3 (a) Derive the Taylor series representation 03
1/(1-z) = ? zn+1 > |z - 1| < 1
(b) State Cauchy Integral formula. Use it to evaluate ? dz C: |z| =1. 04
cz(z2 +8)
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(c) Find Laurent series representation of f(z)= 1/z2(1 +z ) 07
for (i) 0 <|z| <1 (ii) 1<|z|<8 .
OR
Q.3 (a) Determine residue of f(z)= (3z +9)/(z2+9) at z=3i. 03
(b) Find the fixed points of the transformation w = (z—i)/(z+i) . 04
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(c) Evaluate ? (x dx)/(x2+1)3 using residue theorem. 07
Q.4 (a) Show that 1+?=E 03
(b) Find f(0.12)& f(0.26) by appropriate interpolation formula from following 04
table
| X | 0.10 | 0.15 | 0.20 | 0.25 | 0.30 |
|---|---|---|---|---|---|
| f(x) | 0.1003 | 0.1511 | 0.2027 | 0.2553 | 0.3093 |
(c) Determine images of Vertical and Horizontal lines under the transformation 07
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w=ez.
OR
Q.4 (a) Using Lagrange’s formula, express the function 2/((x-1)(x-2)(x-3)) as a sum 03
of partial fractions.
(b) Find interpolating polynomial using Newton’s divided difference formula from 04
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following table
| X | 0 | 1 | 4 | 5 | 7 |
|---|---|---|---|---|---|
| f(x) | -6 | -3 | 138 | 369 | 1611 |
(c) Show that a function f(z)=u(x,y)+iv(x,y)is analytic in a domain D if and 07
only if v is a harmonic conjugate of u.
Q.5 (a) Use Newton-Raphson method to find positive root of sinx =1-x correct to 03
three decimal places.
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(b) Use power method to find largest eigen value and corresponding eigen vector 04
of [1 2; 2 4] correct to four decimal places.
(c) Apply Runge-Kutta fourth-order method to find y(0.2). Given that 07
dy/dx =y—x where y(0)=2 and h=0.1.
OR
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Q.5 (a) Use Secant method to find a positive root of the equation x3 +x—1= 0 correct 03
to three decimal places.
(b) Given that dy/dx =x2 +y, y(0)=1. Find y(0.1) using Modified Euler’s 04
method with h= 0.05 correct to three decimal places.
(c) Solve the following liner system 07
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10x+2y+z=9
2x+20y-2z=-44
—2x+3y+10z=22
Correct to two decimal places by Gauss-Seidel method.
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