Complex Variables and Numerical Methods — Summer 2018

Separate real and imaginary parts of f ze z . Also prove that it is analytic everywhere.

Write De Moivre’s theorem. Find out cube root of1 i .

Apply Gauss elimination method to solve system of equations:6 x x y   2 y y z   8  z z7  2 2z e

Evaluate  dz, where zln1.53 C Cis the square with vertices (1,0), (0,1), (0-1) and (-1,0)

Verify whether u  x , y   x 2  y 2 is harmonic or not? Also find out its harmonic conjugate.

State Cauchy – Riemann theorem. Write C-Requations in polar form and verify it for f  z   z z07 in polar form.

Evaluate Cz 2 d z where Cis the boundary of the square with vertices (0,0), (1,0), (1,1), (0,1).

Find the radius of convergence of the series 2 n  l z o n g n  303 z 1

Evaluate  dz , if  2  z 1 C Cis the circle of unit radius with center (I) at z  1 and (II) atz1.

Evaluate following integrals using residue:2 2z 3 (1)  dz, where C: z 1.6 zz1z2 C  3x2 (2)  dx  xx4  x 2 9 

Find series expression for f  z   t a n  1 z at z  0

Find the bilinear transformation which maps the points z  1 , i ,  1 in to the points w  i , 0 , 104

Find all Taylor and Laurent series of f  z   z 2  z3  z3  with center 0.

Evaluate  x2 l o g x d x using Trapezoidal rule with step size h  1

Apply Lagrange’s interpolation formula to evaluate Y ( 3 ) using data given below: X 0 1 2 Y -1 2 7 23

Find the 3th root of the 119 correct up to five decimal places, using Newton Raphson method.

Evaluate  d  t t using Two point Gaussian formula.

Use Bolzano method to find the positive root of x  c o s x correct up to three decimal places. Use initial values 0 .7 2 and 0 . 7 504

Use appropriate Newton’s formula to find the values of Y ( 2 1 ) and Y ( 2 8 ) from the data given below: X 20 23 26 29 Y 79.8 121.44 175.5 243.6

Apply Gauss Jacobi method to solve system of linear equation as under: x2  x x4  0 y2 y  y 1 3 z z z   3 3

Given that d d y x  3 x  y with y  1   1 . 3 , consider step size h  0 . 1 . Find y  1 .1  and y  1 . 2  using Euler’s method. Also find y  1 . 3 07 using Runge- Kutta method of order 4.

Determine the largest Eigen value and the corresponding eigen vector of the matrix A   44 07 dy

Using Taylor’s Series method solve  x 2 y1,y01. Also find dx y0.03