Complex Variables and Numerical Methods — Summer 2023

(i) Find the real part of ( − 1 − i ) 7 + ( − 1 + i ) 7 [03] (ii) Is Ar g ( z1 z2 ) = Ar g ( z1 ) + Ar g ( z2 ) ? Justify your answer. [04]

(i) Sketch the graph of the setS ={ z−1+2i  2} [03] Does it define a domain? (ii) If f ( x , y ) = = x0 x4 , i2 + f y y ( x2 , , y i f ) = x (2 0 , +0 y ) 2  0 Show that ( x ,y lim )→ (0 , 0 ) f ( x , y ) [04] does not exist.

(i) Check whether f(z) =sin z is analytic function or not. If so, find its derivative. [03] (ii) Show that the real and imag inary parts of an analytic function are harmonic [04] functions.

Evaluate C Re ( z 2 ) d z where c is a boundary of the square with vertices [07] 0, i, 1 + i, 1 in the clockwise direction.

Evaluate C ( z + 1 z ) −2 ( z − 2 ) d z where c is the circle z − i = [07]

 ( 2n ) ! [07] Discuss the convergence of  ( z−3i )n .

( n! )2 n=0 Find also the radius of convergence.

Define the Mobius transformation. Apply it to find the image of z =1 under w = i i − + z z [07]

2 sin z [07] State Cauchy’s residue theorem. Use it to evaluate  dz where Cis

4z2 −1 C z =

Using Schwarz-Christoffel transformation, find the transformation which maps the angular region2 z n  0  a r g  onto the half plane v0 and the points z = 0 and z = 1 [07] onto w=0 and w =1 respectively in the w-plane.

(i) Define forward, backward and central difference operators. [03] (ii) Using Newton’ forward interpolation formula, find y at x =1.5 from the data given below x 0 1 2 3 y − 10 − 8 − 8 − [04]440

Explain Gauss-Seidel method for solving a linear system of three equations in three unknowns x, y and z. Apply it to solve 2 x + y + 5 4 z = 1 1 0 , 6 x + 1 5 y + 2 z = 7 2 , 2 7 x + 6 y − z = 8 [07] correct to four decimal places.

3 [07] ( )2

Evaluate  1+ x2 dx by Gaussian quadrature formula for n = −2

State Newton’s divided difference formula. Apply it to find a polynomial of [07] the degree three from the following data: x 1 2 7 y 1 5 5

(i) By power method, determine the largest eigen value of A =  − 11  [03] (ii) Using secant method, find a real root of x 3 − 2 x − 5 = 0 correct to three decimal places starting with x0 = 2 and x1 = [04]

Using Euler’s formula, solve y for x = 0.1 from [07] dy = x + y +xy, y(0)=1and h=0.02 dx

(i) Apply residue theorem to evaluate  +   2 d s in [03] (ii) Use bisection method to find first four iterations for [04] x3 −4x−9=0 taking x =2.7050

Using fourth order Runge-Kutta method, find y at x = 1 given that [07] dy y−x = , y(0)=1and h=0.5 dx y+x