MA301BS: MATHEMATICS  IV
(Complex Variables and Fourier Analysis)
B.Tech. II Year I Sem. L T P C
4 1 0 4
Prerequisites: Foundation course (No Prerequisites).
Course Objectives: To learn
 differentiation and integration of complex valued functions
 evaluation of integrals using Cauchy’s integral formula
 Laurent’s series expansion of complex functions
 evaluation of integrals using Residue theorem
 express a periodic function by Fourier series and a nonperiodic function by Fourier transform
 to analyze the displacements of one dimensional wave and distribution of one dimensional heat equation
Course Outcomes: After learning the contents of this paper the student must be able to:
 analyze the complex functions with reference to their analyticity, integration using Cauchy’s integral theorem
 find the Taylor’s and Laurent’s series expansion of complex functions
 the bilinear transformation
 express any periodic function in term of sines and cosines
 express a nonperiodic function as integral representation
 analyze one dimensional wave and heat equation
UNIT – I
Functions of a complex variable: Introduction, Continuity, Differentiability, Analyticity, properties, Cauchy, Riemann equations in Cartesian and polar coordinates. Harmonic and conjugate harmonic functionsMilneThompson method
UNIT  II
Complex integration: Line integral, Cauchy’s integral theorem, Cauchy’s integral formula, and Generalized Cauchy’s integral formula, Power series: Taylor’s series Laurent series, Singular points, isolated singular points, pole of order m – essential singularity, Residue, Cauchy Residue theorem (Without proof).
UNIT – III
Evaluation of Integrals: Types of real integrals:
a) Improper real integrals
(b)
Bilinear transformation fixed point cross ratio properties invariance of circles.
UNIT – IV
Fourier series and Transforms: Introduction, Periodic functions, Fourier series of periodic function, Dirichlet’s conditions, Even and odd functions, Change of interval, Half range sine and cosine series.
Fourier integral theorem (without proof), Fourier sine and cosine integrals, sine and cosine, transforms, properties, inverse transforms, Finite Fourier transforms.
UNIT – V
Applications of PDE: Classification of second order partial differential equations, method of separation of variables, Solution of one dimensional wave and heat equations.
TEXT BOOKS:
 A first course in complex analysis with applications by Dennis G. Zill and Patrick Shanahan, Johns and Bartlett Publishers.
 Higher Engineering Mathematics by Dr. B. S. Grewal, Khanna Publishers.
 Advanced engineering Mathematics with MATLAB by Dean G. Duffy
REFERENCES:
 Fundamentals of Complex Analysis by Saff, E. B. and A. D. Snider, Pearson.
 Advanced Engineering Mathematics by Louis C. Barrett, McGraw Hill.

CreatedMay 27, 2017

UpdatedMay 27, 2017

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