Complex Variables and Numerical Methods — Winter 2021

State De Moivre’s theorem. Find the roots of the equation z4 +1 = 0.

Find the real and imaginary parts of f(z) = sinz and check whether Cauchy Riemann equations are satisfied at all points. Hence find f′(z).

( i ) Use Lagrange’s interpolation formula to estimate the value of y at x = from the following data: x 3 7 9 11 y 168 120 72 48 ( ii ) Rearrange the following system of equations in diagonally dominant form and then solve the new system using Gauss Siedel Method: x+20 y+z = 44, x+y+20 z = 63, 20 x+y+z = 25.

Perform four iterations of Bisection Method to find a root of f(x) = x3 −4x +1 = 0.

Find the bilinear transformation which maps z = 0,1,∞ into the points w = i,1,−i.

∞ x2dx Evaluate ∫ using contour integration. −∞(x2+1)(x2+4)

( i ) Show that the function f(x,y) = 2xy+i(x2 −y2) is differentiable at a single point z = 0. Hence find f′(0). ( ii ) Prove that u(x,y) = x2 −y2 +e−2 xcos2y is harmonic and find its harmonic conjugate.

z Evaluate ∮ dz if Cis the circle |z+i| = 1. Cz2+1

Find the real and imaginary part of (1+i)i.

Evaluate ∫ 1+i (x2 +iy)dz along the path (i) y = x, (ii) y = x2. Is the line0 integral independent of the path?

sinz−z Find the type of singularity of the function f(z) = . z3

Find and sketch the image of |z+1| = 1 under the transformation w = 1 . z

Expand f(z) = 1 in the Laurent series valid in the regions (z+1)(z+3)

|z| > 3 (ii) 1 < |z| < 3.

If f(z) and f ( z ) are both analytic, then prove that f(z) is constant.

Use modified Euler’s method to find y(0.2) given that dy = 1−y, y(0) = 0, ℎ = 0.1. dx

( i ) Find a root of the equation x = cos x using Newton-Raphson Method correct up to four decimal places. Take x = 1.0 ( ii ) Use Newton’s divided difference formula to determine the polynomial which fits with the following data: x 0 1 2 4 5 f(x) 1 14 15 5 6 19

If f(z) is analytic, then prove that 𝜕2 𝜕2 ( + )|Re f(z)|2 = 2 |f′(z)|2. 𝜕x2 𝜕y2

Solve the following system of equations using Gauss Elimination Method: x+y+z = 3, x−y+z = 1, −x +y+2z = 2.

( i ) Find the real root of the equation 2x −logx = 7 in (3.5,4.0) using False10 Position Method correct to three decimal places. ( ii ) The population (in thousands) of a town is given below. Estimate the population for the year 1975 using interpolation. Year 1971 1981 1991 2001 2011 Population 46 66 81 93 101

Evaluate ∫ 1 xe−xdx by using Gauss quadrature formula with n = 3.

Use fourth order Runge-Kutta Method to compute y(0.1) and y(0.2) given that dy 4x = y+ , y(0) = 1, ℎ = 0.1. dx y

( i ) Evaluate ∫ 5.2 log x dx using Simpson’s 3⁄8 rule. Take ℎ = 0.2.4 ( ii ) Find the dominant Eigenvalue of A = [ ] by Power Method and the13 corresponding Eigenvector.

The following table gives the velocity v of a particle at time t: t (sec.) 0 2 4 6 8 10 v (m /sec.) 4 6 16 34 60 94 135 Find the distance moved by the particle in 12 seconds.

Find an approximate value of y when x = 0.1 using modified Euler’s method if dy = x2 +y, y(0) = 1, ℎ = 0.05. dx Perform two corrections at every step.

Prove the following identities: ∆ ∇

∆+∇= − ∇ ∆12 (ii) 1+𝜇2𝛿2 = (1+ 𝛿2)2 (iii) 𝜇𝛿 = ∆E−1 + ∆.22