1
Seat No.: Enrolment
GUJARAT TECHNOLOGICAL UNIVERSITY
BE SEMESTER-IV • EXAMINATION WINTER 2016
Subject Code: 2141905 Date: 18/11/2016
Subject Name: Complex Variables and Numerical Methods
Time: 02:30 PM to 05:30 PM Total Marks: 70
Instructions:
1. Attempt all questions.
2. Make suitable assumptions wherever necessary.
3. Figures to the right indicate full marks.
MARKS
Q.1 Short Questions 14
1
Find the principle angle of .
3
1
i
2 Define analytic function.
3 Write necessary condition for differentiability of f .
4
Find the real part of at point 1 .
2
1
i
z i
f z


5 Define harmonic function.
6 Separate real and imaginary part of 2 f z .
7
Classify the singular point z=0 for the function
4 2 4
1

z z
f z


8 Show that E
9
Check whether
z
w
1
is conformal mapping or not.
10 Write trapezoidal rule.
11 When does Newton- Raphson method fail to find root of equation.
12 Can we use Gauss- Seidel method to solve the system of linear equation
3x y 2z 8,2x 3y z x 2y 3z 6 .
13 Express 2 y x in factorial notation.
14
Find for
x
f x
1
.
Q.2
Check whether functions 2
3
f z is analytic or not. Also find
derivative of f
03
Find the bilinear transformation which maps z into the points
w
04

Evaluate the Cauchy's principle value of

9 2 2 x x
dx
.
07
OR
Show that e y x cos is harmonic function. Also find harmonic
conjugate of .
07
Q.3
Using Cauchy integral formula, Evaluate


C
dz
z z
z
2 4
3 2
2
2
where C is
a circle z 2 2 .
03
2
Find and plot all the roots of 8 . 3 i 04
Evaluate
C
z dz 2 where C is taken along triangle in z-plane having
vertices z z taken in counter clockwise sence.
07
OR
Q.3
Expend
1



z
e
f z
z
about z=1. Also classify singular point z=1.
03

Discuss the continuity of






0 0
0

2
z
z
z
z
f z at z=0.
04

Expand

1


z z
f z valid for the region
z 2 2 z 4 z 4 .
07
Q.4 Solve by Gauss Elimination method
x 2y z 2x 3y 3z 3x y 2z
03
Derive iterative formula to find N . Use this formula to find 28. 04
Determine the polynomial by Newton's forward difference formula
from the following table:
0 1 2 3 4
-10 40
Also find y when x=1.5.
07
OR
Q.4 Using Euler's method, find an approximate value of corresponding to
x=0.3, given that 1.


y
y x
y x
dx
dy
Take h 0.1.
03
Find a real root of the equation 2 5 0 3 x x using secant method
correct to three decimal places taking 2 and 3. 0 1 x x
04
Determine the interpolating polynomial of degree three using
Lagrange's interpolation for the table below:
0 1 3
2 1 0
Also find the value of y when x=2.
07
Q.5 The velocity v of a particle at a distant s from a point on its path is
given by the following table:
s 0 10 20 30 40 50 60
v(ft/sec) 47 58 64 65 61 52 38
Estimate the time taken to travel 60 ft. using Simpson's
8
3
th rule.
03
Using Runge- Kutta method of fourth order to calculate y(0.2) given
that x 1
dx
dy
taking h=0.1.
04
Solve by Gauss-Seidel method
10x y z x z x y 6.
07
OR
Q.5
Evaluate
1
0 1
1
dx
x
by Gaussian formula with two point.
03
Find a positive root of 2 0 x x e by the method of False position. 04
3

Find the dominant eigen value of




3 4
1 2
A by power method and
hence find the other eigen value also.
07