Complex Variables and Numerical Methods — Summer 2020

Verify Cauchy-Riemann equation for f(z)cosxcosh yisinxsinh y .

Find all cube roots of complex number 8i.04

Evaluate  dx with h1 by (i) Trapezoidal Rule (ii) Simpson’s Rule 1x2 30 (iii) Simpson’s Rule.8

Find the principal value of 1i4i. z2

Using Parametric representation of C, evaluate  dz; Cis the circle04 Cz z 2ei02

Show that u(x,y)2xx33xy2is harmonic function and find harmonic07 conjugate v(x,y).

 2 z 

; z 0 For f(z) , z  0 ; z 0 Show that C-Requation is satisfied at origin but f'(0) does not exist.

Derive the Taylor series representation 1  zin  ; zi  2. 1z (1i)n1 n0 cosz

State Cauchy Integral formula. Use it to evaluate  dz; C: z 1.04 Cz(z2 8)1

Find Laurent series representation of f(z) z  1z2 for (i) 0 < z 1 (ii) 1 z  .

3z32

Determine residue of f(z) at z3i. z2 9 z1

Find the fixed points of the transformation w .04 z11  x2dx

Evaluate  using residues.  x2 1  x2 4 0

Show that 1ehD

Find f(0.12)& f(0.26)by appropriate interpolation formula from following04 table x 0.10 0.15 0.20 0.25 0.30 f (x) 0.1003 0.1511 0.2027 0.2553 0.3093

Determine images of Vertical and Horizontal lines under the transformation wez.

3x2 x1

Using Lagrange’s formula, express the function as a sum (x1)(x2)(x3) of partial fractions.

Find interpolating polynomial using Newton’s divided difference formula from following table x 0 1 4 5 f (x) 6 3 138 369 1611

Show that a function f(z)u(x,y)iv(x,y)is analytic in a domain Dif and only if vis a harmonic conjugate of u.

Use Newton-Raphson method to find positive root of sinx1x correct to three decimal places.

Use power method to find largest eigen value and corresponding eigen vector 1 2 of correct to four decimal places.   3 4

Apply Runge-Kutta fourth-order method to find y(0.2). Given that dy  yx where y(0)2and h0.1 . dx

Use Secant method to find a positive root of the equation x3 x10correct to three decimal places. dy

Given that  x2  y; y(0)1. Find y(0.1) using Modified Euler’s dx method with h0.05correct to three decimal places.

Solve the following liner system 10x2yz 9 2x20y2z 44 2x3y10z 22 Correct to two decimal places by Gauss-Seidel method.