G V P College of Engineering (Autonomous)               2015

54 EEE

SPECIAL FUNCTIONS ANDCOMPLEX VARIABLES

Course Code: 15BM1104 L T P C3 0 0 3

Pre requisites:

1. Basic Knowledge in evaluation of definite integrals.

2. Calculus of functions of real variables.

Course Outcomes:

At the end of the Course, Student will be able to:

CO 1 Compute improper integrals using beta and gamma functionsand discuss the properties of the Legendre polynomial.

CO 2 Discuss various properties of the Bessel’s function.

CO 3 Examine continuity and analyticity of various complex valuedfunctions.

CO 4 Determine Taylor’s and Laurent’s series of a complex functionand use residue theorem to evaluate certain real definiteintegrals.

CO 5 Transform various regions using conformal mappings.

UNIT-I: (10 Lectures)SPECIAL FUNCTIONS-1 (BETA, GAMMA AND LEGENDREFUNCTIONS)

Beta-function, Gamma function, Relation between Beta and Gammafunctions, Series solution of Legendre’s equation, Legendre’s function,Rodrigue’s formula, Legendre polynomials, Generating function,Recurrence formulae(7.14-7.16, 16.13-16.16)

G V P College of Engineering (Autonomous)               2015

55EEE

UNIT-II: (10 Lectures)SPECIAL FUNCTIONS-2 (BESSEL FUNCTION)

Bessel’s equation, Bessel’s function, Recurrence formulae for Bessel

function, Expansions for and , value of , Generating

function for , Orthogonality of Bessel functions.

(16.5-16.9, 16.11(1))

UNIT-III: (10 Lectures)

FUNCTIONS OF A COMPLEX VARIABLE:

Complex function, Real and Imaginary parts of Complex function,Limit, Continuity and Derivative of a Complex function, Cauchy-Riemann equations, Analytic function, entire function, singular point,conjugate function, Cauchy-Riemann equations in polar form,Harmonic functions, Milne-Thomson method, Simple applicationsto flow problems.

COMPLEX INTEGRATION:

Line integral of a complex function, Cauchy’s theorem (onlystatement), Cauchy’s Integral Formula (without proof)

(19.7, 19.12, 20.2-20.6, 20.12-20.14)

UNIT-IV: (10 Lectures)

SERIES OF COMPLEX TERMS:

Absolutely convergent and uniformly convergent series of complexterms, Radius of convergence, Taylor’s series, Maclaurin’s seriesexpansion, Laurent’s series (without proofs). Zeros of an analyticfunction, Singularities of a complex function, Isolated singularity,Removable singularity, Poles, pole of order m, simple pole, Essentialsingularity, Residues: Residue theorem, Calculation of residues,Residue at a pole of order m, Evaluation of real definite integrals:Integration around the unit circle, Integration around a semi circle

( 20.16 - 20.20 (a), (b))

G V P College of Engineering (Autonomous)               2015

56 EEE

UNIT-V: (10 Lectures)

CONFORMAL TRANSFORMATION

Standard transformations: Translation, Magnification and rotation,Inversion and reflection, Bilinear transformation and its Properties,Conformal transformation, critical point, fixed points of atransformation, Special Conformal transformations: , ,

, , .( excluding hyperbolic

functions)

(20.8-20.10)

TEXT BOOK:

1. Dr. B.S.Grewal, “Higher engineering mathematics”, 42ndedition, Khanna publishers, 2012.

REFERENCE BOOKS:

1. Kreyszig E, “Advanced Engineering Mathematics”, 8thEdition. John Wiley, Singapore, 2001.

2. Glyn James, “Advanced Modern Engineering Mathematics”,3rd edition, Pearson, 2004.