# Complex Variables and Partial Differential Equations(3130005)

BE | Semester 3
Summer - 2020|27-10-2020
Total Marks 70

## Q1 (a) Find the analytic function

Unit : Complex variable functions |  Topic : Harmonic functions

## Q1 (b) Find the roots of the equation, .

Unit : Complex variable functions |  Topic : Powers and Roots

## Q1 (c ( i )) Determine & sketch the image of under the transformation .

Unit : Complex variable functions |  Topic : Conformal mappings

## Q1 (c ( ii )) Find the real and imaginary parts of .

Unit : Complex variable functions |  Topic : Conformal mappings

## Q2 (a) Evaluate along the parabola from to .

Unit : Complex Variable - Integration

## Q2 (b) Determine the Mobius transformation which maps into .

Unit : Complex variable functions |  Topic : Mobius transformations and their properties

## Q2 (c(i)) Evaluate  ,where .

Unit : Complex Variable - Integration |  Topic : Cauchy-Goursat theorem

## Q2 (c(ii)) Find the radius of convergence of  .

Unit : Complex Variable - Integration |  Topic : Power Series

## Q2 (c( i )) Find the fourth roots of -1.

Unit : Complex variable functions |  Topic : Powers and Roots

## Q3 (b) For, , find the residue at .

Unit : Laurentâ€™s series |  Topic : Residues

## Q3 (c) Expand  valid for the region and

Unit : Laurentâ€™s series |  Topic : Laurentâ€™s series

## Q3 (b) Evaluate using Cauchy’s residue theorem , Where, .

Unit : Laurentâ€™s series |  Topic : Cauchy Residue theorem

## Q3 (c) Expand  valid for the region and

Unit : Laurentâ€™s series |  Topic : Cauchy Residue theorem

## Q4 (b) Derive partial differential equation by eliminating the arbitrary constants and from .

Unit : First order partial differential equations

## Q4 (c ( ii )) Find the complete integral of .

Unit : First order partial differential equations |  Topic : Solutions of first order nonlinear PDEs

## Q4 (b) Form the partial differential equation by eliminating arbitrary function $\varnothing$ from .

Unit : First order partial differential equations

## Q5 (c) Find the solution of wave equation with initial and boundary condition

Unit : HIgher order partial differential equations and applications |  Topic : Modeling and solution of the Wave equations

## Q5 (c) Find the solution of  together with the initial and boundary conditions

Unit : HIgher order partial differential equations and applications |  Topic : Modeling and solution of the Heat equations