Complex Variables and Numerical Methods — Winter 2020

o Q.6. 2. Q.7 is compulsory. 3. Make suitable assumptions wherever necessary. 4. Figures to the right indicate full marks.

Separate real and imaginary parts of f(z) = e(z+2), and also prove that it is analytic everywhere.

Use De Moiver’s theorem and find 4th root of unity in the complex plane.

Use Gauss-Jacobi method to determine roots of the following equations 20x +y−2z = 17 3x +20y−z = −18 2x −3y+20z = 25

Evaluate the following integral along the curve z(t) = t+it2 2+4i ∫ Re(z)dz0

Evaluate ∮ cos𝜋z dz where Cis the circle z−1 1) |z| = 2 2) |z| = 1/2

Verify that u = x2 −y2 −y is harmonic in the whole complex plane and finds it’s conjugate harmonic function v.

𝜋 Obtain the Taylor’s series of f(z) = sinz in powers of (z− ).4

Find the center and radius of convergence of the power series ∑∞ (n+2i)nzn. n=0

Find the Laurent’s series expansion of f(z) = 1 in the region (z+1)(z−2) 1) 1 < |z| < 2 2) |z| >

Find the Maclaurin’s series of f(z)=sin2z

Find all values of z such that ez = 1+i

Evaluate ∮ cosz dz counterclockwise around C: |z| = 5 ⁄ z2−4

Use Bisection method to find the real root of x3 −4x −9 = 0 . (Do 4 iterations)

Using Newton’s divided difference interpolation formula, compute f(10.5) from the following data: x 10 11 13 17 f(x) 2.3026 2.3979 2.5649 2.83321

Use Simpson’s 3/8 rule and evaluate the following integral taking n=6, and hence calculate log 2. Also, find the error involved in the pross. e3 dx ∫ 1+x0

Approximate the root of the equation ex −2cosx = 0, by three iterations of Newton Raphson method, taking initial approximation as x = 2.0

Find an approximate value of f(3.6) using Newton’s backward difference formula from the following data: x 0 1 2 3 f(x) -5 1 9 25 55

Using power method, determine the largest eigenvalue and the 2 −1 0 corresponding eigenvector of the matrix = [−1 2 −1] , taking 0 −1 21 initial eigenvector x = [0].0

Using three point Gaussian formula evaluate the following integral and compare with the exact value.1 dx ∫ 1+x2 −1

Solve the following system of linear equations using Gauss Elimination Method. x+y+z = 9;2x −3y+4z = 13;3x +4y+5z = 40