Complex Variables and Numerical Methods — Winter 2024

Find the real and imaginary part of f(z) = z2 +3z

Determine and sketch the regions in the z-plane represented by 1 < |z+2i| ≤

Define Analytic function. Check whether f(z) = sinz is analytic or not. If analytic, find its derivative.

Is f(z) = z (z ≠ 0) is continuous at origin? |z| = 0 for z = 0

e 3𝜋i Find the principal value of [ (−1−i√3)]2

Verify Cauchy’s integral theorem for f(z) = z2 taken over the boundary of a square with vertices at ±1±i in counter-clockwise direction.

2𝜋 4d𝜃 Using the residue theorem, evaluate ∫ 0 5+4sin𝜃

Define Harmonic function. Show that u(x,y) = x2 −y2is harmonic.

Evaluate ∮ dz where Cis |z+i| = 1, counterclockwise. cz2+1

Expand f(z) = 1 in Laurent’s series valid for (z+1)(z+3)

|z| < 1 (ii) 1 < |z| <

Evaluate ∫ ∞ 3z+2 dz −∞z(z−4)(z2+9)

Find ∮tanz dz, where Cis the circle |z| = 2. c

Find the bilinear transformation that maps the point 0,1,i in z-plane onto the points 1+i,−i,2−i in the w-plane.

1 dx Evaluate ∫ , using trapezoidal rule with h=0.2 0 1+x2

Find a root of the equation x3 −x−11 = 0 using the bisection method up to fourth approximation.

Determine y(12) by Lagrange Interpolation from the following X 11 13 14 18 20 23 Y 25 47 68 82 102 1241

Find the square root of 10 correct to three decimal places, by using Newton-Raphson iteration formula.

1 dx 1 State Simpson’s 3/8 rule and evaluate ∫ taking ℎ = 0 1+x2

Interpolate by means of Gauss’s backward interpolation formula, the sales of a concern for the year 1966, given that Year 1931 1941 1951 1961 1971 1981 Sales(in Lakh Rs.) 12 15 20 27 39 52

dy x−y Use Euler’s method to solve the initial value problem = on [0,3] dx with y(0)=1.

Use power method to find the largest eigen value of the matrix A =42 [ ]. Perform four iterations only.13

Solve the following equations by Gauss-Seidel method. 27x +6y−z = 85 6x +5y+2z = 72 x+y+54z = 110

Using Taylor’s series method, find y(0.1) correct up to four decimal dy 2x places given that = y− , y(0) = 1. dx y

Use Runge – kutta second order method to find the approximate value of y(0.2) given that dy = x −y2 and (0) = 1 and ℎ = 0.1. dx

Solve dy = x+y with y(0) = 1 by Euler’s modified method for x = 0.1 dx correct to four decimal places by taking ℎ = 0.05.