SRI VENKATESWARA COLLEGE OF ENGINEERING
(AUTONOMOUS)
Karakambadi Road, TIRUPATI – 517507
Question Bank
Name of the
EEE
Branch/Course
COMBLEX VRIABLE AND NUMERICAL METHODS
Subject
Subject Code MA23ABS301
Year &S em II & I
10 Marks Questions
Unit – 1
𝑥 3(1+𝑖)−𝑦 3 (1−𝑖)
𝑧≠0
Prove that the function 𝑓(𝑧) defined by𝑓 (𝑧) = { 𝑥 2 +𝑦 2
0 𝑧=0
1
is continuous and Cauchy – Riemann equations are satisfied at the origin, yet 𝑓′(𝑧) does not
exist.
2 Prove that 𝑍 𝑛 ( n is a positive integer) is analytic and hence find its derivative.
𝑝𝑥
Determine p such that the function 𝑓 (𝑧) = 2 log(𝑥 2 + 𝑦 2 ) + 𝑖𝑡𝑎𝑛−1 ( 𝑦 )be an analytic
3 function.
a )Show that the function 𝑓 (𝑧) = 𝑙𝑜𝑔𝑧, 𝑓𝑜𝑟 𝑧 ≠ 0 is analytic.
4
b )Find k such that 𝑓 (𝑥, 𝑦) = 𝑥 3 + 3𝑘𝑥𝑦 2 may be harmonic & find its conjugate.
𝜕2 𝜕2
5 If f(z) is a regular function of z, prove that ( + ) | 𝑓(𝑧)|2 = 4|𝑓′(𝑧)|2
𝜕𝑥 2 𝜕𝑦 2
a) Show that the real and imaginary parts of the function w = log z satisfy the C R
6 equations when z is not zero.
b) Find whether 𝑓 (𝑧) = 𝑠𝑖𝑛𝑥 𝑠𝑖𝑛𝑦 − 𝑖𝑐𝑜𝑠𝑥𝑐𝑜𝑠𝑦 is analytic or not.
Show that 𝑢(𝑥) = 𝑒 2𝑥 (𝑥𝑐𝑜𝑠2𝑦 − 𝑦𝑠𝑖𝑛2𝑦) is harmonic and find f(z) interms of z by
7 using Milne –Thomson’s method.
Show that the function 𝑢(𝑥, 𝑦) = 𝑒 𝑥 𝑐𝑜𝑠𝑦 is harmonic. Determine its harmonic conjugate
8 𝑣(𝑥, 𝑦) and also find the analytic function 𝑓(𝑧) = 𝑢 + 𝑖𝑣
If 𝑓 (𝑧) = 𝑢 + 𝑖𝑣 is an analytic function of z and if 𝑢 − 𝑣 = 𝑒 𝑥 (𝑥𝑐𝑜𝑠𝑦 − 𝑦𝑠𝑖𝑛𝑦) then find
9
f(z) in terms of z.
10 Show that the function 𝑢 = 2log(𝑥 2 + 𝑦 2 ) is harmonic and find its hormonic conjugate.
Unit –2
Verify Cauchy’s theorem for the function 𝑓(𝑧) = 3𝑧 2 + 𝑖𝑧 − 4 if c is the square with vertices
1 at1 ± 𝑖 𝑎𝑛𝑑 − 1 ± 𝑖.
Evaluate ∫𝑐 (𝑦 2 + 2𝑥𝑦)𝑑𝑥 + (𝑥 2 − 2𝑥𝑦)𝑑𝑦where Cis boundary of the region by
2
𝑦 = 𝑥 2 𝑎𝑛𝑑 𝑥 = 𝑦 2
𝑒𝑧
a )Using Cauchy’s integral formula, evaluate ∫𝑐 (𝑧−1)(𝑧−2) 𝑑𝑧 where 𝐶: |𝑧| = 3
3 𝑙𝑜𝑔𝑧 1
b )Using Cauchy’s Integral formula, find∫𝑐 (𝑧−1)3 𝑑𝑧 , 𝐶: |𝑧 − 1| = 2
𝜋
a) Find Taylor’s series of 𝑓(𝑧)=𝑠𝑖𝑛𝑧𝑎𝑏𝑜𝑢𝑡 𝑧 = 4
4 𝑧 2 −1
b ) Obtain the Taylor series to represent the function in the region |𝑧| < 2
(𝑧+2)(𝑧+3)
1
5 Find the Laurent’s expansion of𝑧 2−4𝑧+3 𝑓𝑜𝑟 (𝑖 )1 < |𝑧| < 3 (𝑖𝑖 )|𝑧| < 1 (𝑖𝑖𝑖)|𝑧| > 3
4𝑧+4
Find the Laurent’s expansion of 𝑓 (𝑧) = 𝑧(𝑧+2)(𝑧−3) for (i) 1 ≤ |𝑧| ≤ 2 (ii) |𝑧| ≤ 1
6
4−3𝑧 3
7 Evaluate ∫𝑐 𝑧(𝑧−1)(𝑧−2) 𝑑𝑧 where c is the circle 𝑧 = 2 by using residue theorem.
2
d 2
8
Show that 2 cos
0
3
𝑧−3
Evaluate ∫c 𝑧 2+2𝑧+5 𝑑𝑧 where c is circle given by
9
(i)|𝑧| = 1 (ii)|𝑧 + 1 − 𝑖 | = 2 (iii)|𝑧 + 1 + 𝑖 | = 2 by using Residue theorem
2𝜋 𝑑𝜃 2𝜋
10 Show that∫0 = √𝑎2 , 𝑎 > 𝑏 > 0by method of Residues.
𝑎+𝑏𝑠𝑖𝑛𝜃 −𝑏2
Unit – 3
1 Find a positive root of x x 1 0 correct to two decimal places by Bisection method.
3
2 Find a real root of the equation x log10 x 1.2 , which lies between 2 and 3 by Bisection method.
3 Find the root of equation x3 x 4 0 using Regula -falsi method.
4 Find a root of the equation xe x 2 correct to four decimal places by using Regula -falsi method.
5 Find the real root of the equation cos x 3x 1 correct to four decimal places by using iteration
method
6 Find a root of the equation xe cos x 0 correct to four decimal places by using Newton
x
Raphson method.
7 sin x
Consider the following data for f ( x )
x2
x 0.1 0.2 0.3 0.4 0.5
f(x) 9.9833 4.9696 3.2836 2.4339 1.9177
Calculate f (0.25) and f (0.45) accurately using Newton’s method of interpolation.
8 Consider the following data for f ( x )
x 20 25 30 35 40 45
f(x) 354 332 291 260 231 204
Calculate f (22) and f (42) aaccurately using Newton’s method of interpolation.
9 Given values
x 0 1 2 4 5 6
f(x) 1 14 15 5 6 19
Find, f (3) using Lagrange’s interpolation formula.
10 Using Lagrange’s formula, fit a polynomial to the data
x -1 0 2 3
y -8 3 1 12
Also find y at x 1 .
Unit – 4
1 The population of a certain town ( as obtained from census data) is shown in the following table
Year 1951 1961 1971 1981 1991
Population
19.96 39.65 58.81 77.21 94.61
(in thousands)
Estimate the rate of growth of the population in the year 1961 and 1981.
2 Find the value of cos(1.74) and cos(1.86) from the following table.
x 1.7 1.74 1.78 1.82 1.86
sin x 0.9857 0.9916 0.9781 0.9691 0.9584
3 Find the first and second derivatives at x 1.2 from the following data
x 1.0 1.2 1.4 1.6 1.8 2.0 2.2
y 2.7183 3.3201 4.0552 4.9530 6.0496 7.3891 9.0250
4 dy d2y
For the following data, find and at (a) x 1.1 and (b) x 1.6
dx dx 2
x 1.0 1.1 1.2 1.3 1.4 1.5 1.6
y 7.989 8.403 8.781 9.129 9.451 9.750 10.031
6
5 1
Evaluate 1 x dx
0
by using a) Trapezoidal rule b) Simpson’s 13 rule c) Simpson’s 3 8 rule and
compare the results with its actual value.
6 1
Evaluate
0
1 x 3 dx taking h 0.1 using (i) Trapezoidal rule (ii) Simpson’s 1/3 rule
7 Fit a straight line for the following data by the method of least squares.
x 6 7 7 8 8 8 9 9 10
y 5 5 4 5 4 3 4 3 3
8 Fit a polynomial of second degree to the data points given in the following table by the method of
least squares.
x 1 2 3 4 5 6 7
y 2.3 5.2 9.7 16.5 29.4 35.5 54.4
9 Determine the constants a and b by the method of least squares such that y ae
bx
x 1 5 7 9 12
y 10 15 12 15 21
10 Determine the constants a and b by the method of least squares such that y a(b )
x
x 0 1 2 3 4 5 6 7
y 10 21 35 59 92 200 400 610
Unit – 5
1 Solve y x y, y(0) 1
2
Using Taylor’s series method and compute
y (0.1), y (0.2), y(0.3) and y(0.4) correct to four decimal places.
2 Solve y x y, y (1) 0 , Using Taylor’s series method and compute y (1.1) and y (1.2) .
Compare the numerical solution obtained with exact particular solution.
3 dy
Solve y x 2 , y 0 1 by using Picard’s method up to the third approximation. Hence find the
dx
values of y (0.1) and y (0.2) .
4 dy
Solve 1 xy, y 0 1 by using Picard’s method up to the third approximation. Hence find
dx
the values of y (0.1) and y (0.2) .
5 dy
Solve x y, y 0 1 by using Euler’s method and find the value of y (1) , with step size
dx
h 0.2 .
6 dy
Solve 1 2 xy, y 0 0 by using Euler’s method and find the value of y (1) , with step size
dx
h 0.25 .
7 dy
Solve x y, y 0 1 by using modified Euler’s method and find the values of
dx
y (0.1) and y (0.2)
8 Given
dy
e x y, y 0 0 compute y(0.2), y(0.4) Using modified Euler’s method.
dx
9 Find y (0.1) and y (0.2) using Runge-Kutta 4th order formula given that y x 2 y, y(0) 1
dy yx
Find y (0.1) and y (0.2) using Runge-Kutta 4th order formula given that dx y x and y (0)=1.
10
2 Marks Questions
Unit - 1
Define analytic function with an example.
1
State Cauchy – Riemann equations in Cartesian and Polar form
2
(𝑧−2)
Find where the function 𝑓(𝑧) = (𝑧+1)(𝑧 2)fails to be analytic
3
4 Show that 𝑓 (𝑧) = 𝑠𝑖𝑛𝑥𝑠𝑖𝑛𝑦 − 𝑖𝑐𝑜𝑥𝑐𝑜𝑠𝑦 is analytic.
2−𝑦 2
5 Prove that 𝑢 = 𝑒 𝑥 is not harmonic function
Unit-2
State Cauchy’s integral theorem
1
𝑙𝑜𝑔𝑧
Evaluate∫c (𝑧−1)3 𝑑𝑧 where c is |𝑧 − 1| = 1/2
2
𝑧 2+𝑧
3 Identify simple poles of 𝑓 (𝑧) =
𝑧 2 (𝑧−2)(𝑧−3)
4 Expand 𝑒 𝑧 as a Taylor’s series about 𝑧 = 1
1−𝑒 𝑧
5 Find the Residue of 𝑓(𝑧) = at z 0
𝑧4
Unit – 3
If first two approximations of x x 4 0 by Bisection method are x0 1 and x1 2 , then
3
1
find x3 .
2 State Newton’s iteration formula for N.
If the root of the equation xe x 3 lies between x0 1 and x1 2 then find x2 by Regula falsi
3
method.
4 State Newton’s forward and Backward interpolation formulas.
Find the unique polynomial of f(x) of degree 2 or less such that
5 f (1) 0, f (3) 27 and f (4) 64 by using Lagrange interpolation.
Unit – 4
1 State the first and second derivatives by Newton’s interpolation at x x0 .
If f (0) 1, f (0.2) 0.96, f (0.4) 0.86, f (0.6) 0.73, f (0.8) 0.6 and f (1) 0.5 , then by
1
2
Trapezoidal rule f ( x)dx
0
3 State Simpson’s 3/8 rule.
4 If x 15, y
i i 30, xi yi 110, xi2 55 and y a0 a1 x then a1
5 State normal equations for a second-degree polynomial in least square method.
Unit – 5
dy
1 Write the Taylor series for solution of the equation f ( x, y ), y ( x0 ) y0
dx
2 If y x 2 y 2 , and y (0) 0 , then by Picard’s method the value of y (1) ( x) is
3 If y y x , and y (0) 2, h 0.2 , then by Euler’s method the value of y1
4 If y x y 2 , y(0) 1 and h 0.1 , then the value of k2 by fourth order Runge-Kutta method.
5 State the formula for the fourth order Runge-Kutta method.
