University of Mumbai
Class: S.E. Branch:Instrumentation Semester: III
Subject: Engineering Mathematics-III (Abbreviated as EM-III)
Periods per Week(60 min. each)

Lecture

05
Practical
Tutorial
Hours Marks
Evaluation System

Theory

05 100
Practical/Oral

Oral

Term Work

Total

05 100

 all syllabus in single    Click here to download all in PDF

Module

Contents

Hours
1 Laplace TransformFunctions of bounded variationsLaplace Transforms of 1, tn, eat, sin at, cos at, sinh at, cosh at, erf(t)  Linear property of L.T .First shifting theorem   Secondshifting  theorem    L{tn    f(t)},  L{f(t)/t},  L{ f(u)du},  L{dn/dtn f(t)}.Change of  scale  property of  L.T.  Unit  step  function  , Heavyside, Dirac delta    functions, Periodic functions and their Laplace Transforms.

Inverse Laplace Transforms

Evaluation    of    inverse     L.T.,    partial    fractions    method,

convolution theorem.

Applications  to  solve  initial  and  boundary value  problems involving ordinary diff. Equation with one dependant variable.

20
2 Complex Variables.Functions of complex variables, continuity and derivability ofa function, analytic functions, necessary condition for f(z) to be analytic,   sufficient   condition   (without   proof),   Cauchy   – Riemann conditions  in   polar forms. Analytical and Milne – Thomson method to find analytic functions f(z) = u + iv where (i)  u  is  given  (ii)  v  is  given  (iii)  u+v       (iv)  u-v  is  given. Harmonic functions and orthogonal trajectories.Mapping

Conformal  mapping,  Bilinear  mapping,  fixed  points  and

standard  transformation,  inversion,  reflection,  rotation  and magnification.

Line  Integral  of  function  of  complex  variable,  Cauchy’s theorem for analytical function (with proof), Cauchy’s Goursat theorem (without proof),  properties of line integral, Cauchy’s

30

 

 

Integral formula and deduction.Singularities and poles: Taylor’s and Laurent’s development (without   proof),   residue   at   isolated   singularity   and   it’s evaluation.Residue theorem application to evaluate real integrals of type2π                                                +∞

f (cosθ , sin θ )dθ and   f ( x)dx

0                                                                         −∞

3 Fourier seriesOrthogonality  &  orthogonal  functions,  Expression  for  thefunction  in  a  series  of  orthogonal  functions,  Dirichlet’s conditions, Fourier series of periodic functions with period 2pi or  2l.  (Derivation  of  fourier  coefficients  a0,  an,  bn  is  not expected)  Dirichlet’s  theorem  Even  &  Odd  functions.  Half range sine & cosine expressions Parsaval’s identities (without proof)Complex  form  of  Fourier  Series:  Fourier  transform  & Fourier integral in detail 25

Theory Examination:

1.         Question paper will comprise total 7 questions of 20 marks each.

2.         Only 5 questions need to be attempted.

3.         Q.1 will be compulsory covering entire syllabus.

4.         Remaining questions will be of mixed nature.

5.         In  question  paper  weightage  of  each  module  will  be  proportional  to  the number of respective lecture hours as mentioned in the syllabus.

 

Books Recommended:

1. Wartikar P.N. / Wartikar J. N., Textbook of Applied Mathematics, Pune Vidyarthi

Griha Prakashan, 1981.

2. Churchil, Coplex variables, Mc Graw Hill.

3. Shantinarayan, Theory of function Complex Variable, S. Chand & co.

4. Shastri S.S., Engineering Mathematics, Prentice Hall.

 all syllabus in single    Click here to download all in PDF