Complex Variables and Numerical Methods — Winter 2018

Find the roots of the equation

Show that is differentiable only at

Solve the following system of equation by Gauss-Seidal method correct to three decimal places.

03 Evaluate along the line joining the points

Determine the mobius transformation that maps onto respectively.

Prove that the roots of unity are in geometric progression. Also show that their sum is zero.

Verify that C-Requation are satisfied at for the function

03 Evaluate

Find the radius of convergence of

07 Using the residue theorem, evaluate

03 Expand as a Taylor’s series about the point

Check whether is analytic or not. If analytic find its derivative.

07 Evaluate counter clockwise around C, where Cis1

Using Newton’s forward formula , find the value of if x 1 1.4 1.8 2.2 f(x) 3.49 4.82 5.96 6.5

Find the Lagrange interpolating polynomial from the following data x 0 1 4 f(x) 1 3 24 39

Find a root of +7 = 0 correct to three decimal places between by Newton-Raphson method.

Solve the system of equation by Gauss elimination method.

Compute from the following values using Newton’s Divided difference formula x 4 5 7 10 11 13 f(x) 48 100 294 900 1210 2028

07 Evaluate taking and using Simpson’s rule. Hence obtain approximate value of

03 Evaluate

Use power method to find the largest of Eigen values of the matrix

Use Euler’s method to obtain an approximate value of for the differential equation

Prove that

04 Evaluate I = by one point, two point and three point Gaussian formula.

Determine correct upto four decimal places by fourth order Runge-Kutta method from