Separate real and imaginary parts of coshz
[3 marks]Show that f(z) = z3 is analytic everywhere.
[4 marks]After applying partial pivoting to the following system of equations, use Gauss Jacobi method up to 3 iterations and find the approximate solution of the following system of equations. x+2y+5z = x+4y+2z = 5x +2y+z =
[12 marks]Evaluate∫ 𝜋e𝜋z dz, wh5exre+ C2 iys t+hez l=ine1 s2e gment joining the points C 1,1+i.
[3 marks]Find the Bilinear Transformation which maps the points z = 1,z =12 0 and z = −1 of z−plane onto the points w = i,w =312 ∞ and w = 1 of w −plane.3
[4 marks]Show that the transformation w = iz+i maps the half plane x > 0 onto the half plane v > 1
[7 marks]Show that u(x,y) = 2x −x3 +3xy2 is harmonic in some domain and find a harmonic conjugate v(x,y).
[7 marks]Show that when z ≠ 0, ez 1 1 1 z z2 = + + + + + ⋯ z2 z2 z 2! 3! 4!
[3 marks]Find the center and radius of convergence of the power series n2 ∑∞ (1+ 1 ) zn n=1 n
[4 marks]Find the Laurent’s series of the function f(z) = 1 (z−2)(z−3) in the regions
[7 marks]2 < |z| < 3 (b) |z| >
[3 marks]Find the ∫ g(z)dz where, C: |z−i| = 2 and g(z) =1 .03 C (z−2i)
[ marks]Find the ∫ g(z)dz , where C: |z| = 2 and g(z) = 1 C (z2−5z+4)
[4 marks]Using residue theorem, evaluate ∮ 2z+6 dz, where C: |z−i| = z2+4
[7 marks]Find a root of the equation x3 −4x−9 = 0 using the bisection method in 3 iterations.1
[3 marks]Expand f(z) = 1−cosz in Laurent’s series about z = 0 and identify z2 the singularity.
[4 marks]Use Newton’s backward interpolation formula to find the polynomial which takes the following values: y(0) = 1, y(1) = 0, y(2) = 1,y(3) = And hence find y(1.2)
[10 marks]Find a root of x4 −x3 +10x +7 = 0 using Newton’s Raphson method taking initial point -1.5.
[3 marks]Solve the following system of equations by Gauss Elimination method. x+y+z =9 2x −3y+4z = 13 3x +4y+5z = 40
[4 marks]Evaluate ∫ 6 dx using Simpson’s 1 ⁄ rd rule, taking n = 6. Hence, 0 1+x2 find tan−16.
[3 marks]Use Euler’s method solve for y at x = 0.1 from dy = x+y+xy,y(0) = dx 1 in 3 steps
[3 marks]1 dt Evaluate ∫ by Gaussian formula for n=3. 0 1+t
[4 marks]Using the power method find the largest Eigen value for the matrix 1 −3 A = [4 4 −1]635
[2 marks]Prove that E = eℎD
[3 marks]Find a positive root of xex −2 = 0 correct to two places of decimals by the method of False position, taking initial interval (0, 1).
[4 marks]Apply Runge-Kutta fourth order method to find an approximate value of y when x = 0.2 in steps of 0.1 if dy = x+y2, given that y = dx 1 when x = 0.
[7 marks]