Complex Variables and Numerical Methods — Winter 2023

If Zand Zbe two complex numbers, show that ( Z1 + Z2 ) 2 + ( Z1 − Z2 ) 2 = 2  Z1 2 + Z1 2 03

Si m p l i f y ( ( c c o o s s3 i i s s i i n n3 ) )2 ( ( c c o o s s5 i i s s i n i n ) )13         − + − −04

Evaluate ( 1 + i ) 9 0 + ( 1 − i ) 9 0

If a r g ( z 1 )6 a n d a r g ( z 1 )3   + = − =03 , then find the complex number Z.

If 1+2i is a root of the equation Z4 −3Z3 +8Z2 −7Z +5=0, then find all other roots.

Solve z 4 + 1 = 0 07 and locate the roots in argand diagram.

Find the real and imaginary part of5 + −2 i i

Define Harmonic function and determine whether the function ( x , y ) e x c o s y  =04 is harmonic?

Using the residue theorem, evaluate2 d 3 s i n     −07

Determine the analytics function whose real part is u(x,y)= x2 − y2 .

Find the bilinear transformation that maps respectively the points i,1,−i in z - plane onto the points −i,1,i in w – plane.

 dx Evaluate  (x +1)( x2 +3 ) −

Find the real root of the equation x3 +4x2 −1=0 by bisection method.

Evaluate2 601 + x d x04 taking h = 1 using Simpson’s 1/3 rule.

Compute the value of f(7.5), by using suitable interpolation formula using the following table of data. x 3 4 5 6 7 F(x) 28 65 126 217 344 513

Find the positive root of x = c o s x using Newton’s method correct to 3 decimal places.

State the trapezoidal rule with n = 10 and evaluate 10 e x d x04

Using Lagrange’s interpolation formula, find the value of f (0) for the table given below : x -1 -2 2 F(x) -1 -9 11 69

Using Euler’s method, find y ( 0 . 2 ) given d d y x = y −2 y x , y ( 0 ) = 103 ,take h = 0.1

Solve the following system of equations by Gauss Seidel method. 1 0 x + y + z = 6 , x + 1 0 y + z = 6 , x + y + 1 0 z = 604

Use the Runge-Kutta method to solve d d y x = − x y 2 f o r 0  x  1 , subject to y ( 0 ) = 207 . Use h = 0.25 and find y(0.5) .

Using Taylor’s series method, find correct to four decimal places, the value of y ( 0 . 1 ) , given y ' = x 2 + y 2 and y ( 0 ) = 103 .

Solve the following system of equations by Gauss Seidel method. 20x+2y+ z =30,x−40y+3z =−75,2x− y+10z =30

dy Use the Runge-Kutta method to solve = x+ y, subject to dx y ( 0 ) = . Use h = 0.2 and find y(0.4) .