# Complex Variables and Partial Differential Equations(3130005)

BE | Semester 3
Winter - 2019|26-11-2019
Total Marks 70

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

Unit : Complex variable functions

## Q1 (b) State De-Movire’s formula and hence evaluate

Unit : Complex variable functions |  Topic : Powers and Roots

## Q1 (c) Define harmonic function. Show that  is a harmonic function. Find its harmonic conjugate $\mathrm{v}\left(\mathrm{x},\mathrm{y}\right)$.

Unit : Complex variable functions |  Topic : Finding harmonic conjugate

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

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

## Q2 (c) Expand valid for the region , and $\left(\mathrm{III}\right)\left|\mathrm{z}\right|>2$.

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

## Q2 (c) Find the image of the infinite strips under the transformation . Show the region graphically.

Unit : Complex variable functions |  Topic : Conformal mappings

## Q3 (a) Evaluate ; Where $\mathrm{C}$ is a straight line from .

Unit : Complex Variable - Integration

## Q3 (b) Check whether the following functions are analytic or not at any point,

Unit : Complex variable functions |  Topic : Analytic functions

## Q3 (c) Using residue theorem, evaluate  .

Unit : Laurentâ€™s series |  Topic : Residue Integration of Real Integrals

## Q3 (a) Expand Laurent series of  at and identify the singularity.

Unit : Laurentâ€™s series |  Topic : Singularities

## Q3 (b) If  , is an analytic function,prove that  .

Unit : Complex variable functions |  Topic : Analytic functions

## Q3 (c ( i )) Evaluate the following: ; Where C is the circle .

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

## Q3 (c ( ii )) ; where C is the circle .

Unit : Complex Variable - Integration |  Topic : Cauchy Integral formula

## Q4 (a) Evaluate  along the curve .

Unit : Complex Variable - Integration

## Q4 (c) Solve :  for the condition of heat along rod without radiation subject to the conditions  for and & at for all $\mathrm{x}$.

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

## Q4 (b) Solve  using Charpit's method.

Unit : First order partial differential equations |  Topic : Charpitâ€™s Method

## Q4 (c) Find the general solution of partial differential equation using method of separation of variables.

Unit : HIgher order partial differential equations and applications |  Topic : Separation of variables method to simple problems in Cartesian coordinates

## Q5 (c) A string of length has its ends fixed at and . At time , the string is given a slope defined by , then it is released. Find the deflection of the string at any time $\mathrm{t}$.

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

## Q5 (b) Find the temperature in the thin metal rod of length $\mathrm{L}$ with both the ends insulated and initial temperature is .

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

## Q5 (c) Derive the one dimensional wave equation that governs small vibration of an elastic string. Also state physical assumptions that you make for the system.

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