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SYLLABUS

Unit-I

Complex Variable - Differentiation: Introduction to Functions of Complex Variable - Concept of Limit & Continuity - Differentiation, Cauchy-Riemann Equations, Analytic Functions (Exponential, Trigonometric, Logarithm), Harmonic Functions, Finding Harmonic Conjugate - Construction of Analytic Function by Milne-Thomson Method-Conformal Mappings - Standard and Special Transformations (sin z, ez , cos z, z2 ) Mobius Transformations (Bilinear) and Their Properties.

Unit-II

Complex Variable - Integration: Line Integral - Contour Integration - Cauchy’s Integral Theorem - Cauchy Integral Formula - Liouville’s Theorem (Without Proof) and Maximum-Modulus Theorem (Without Proof): Power Series Expansions: Taylor’s series - Zeros of Analytic Functions - Singularities - Laurent’s series - Residues - Cauchy Residue Theorem (Without Proof) - Evaluation of Definite Integral Involving sine and cosine - Evaluation of Certain Improper Integrals (Around Unit Circle, Semi Circle With f(z) not Having Poles on Real Axis).

Unit-III

Laplace Transforms: Definition : Laplace Transform of Standard Functions, Existence of Laplace Transform, Inverse Transform, First Shifting Theorem, Transforms of Derivatives and Integrals, Unit Step Function, Second Shifting Theorem, Dirac’s Delta Function, Convolution Theorem, Laplace Transform of Periodic Function, Differentiation and Integration of Transform, Solving Initial Value Problems to Ordinary Differential Equations with Constant Coefficients using Laplace Transforms.

Unit-IV

Fourier Series: Determination of Fourier Coefficients (Euler’s) - Dirichlet Conditions for the Existence of Fourier series - Functions Having discontinuity - Fourier Series of Even and Odd Functions - Fourier Series in an Arbitrary Interval - Half-Range Fourier Sine and Cosine Expansions - Typical Wave Forms - Parseval’s Formula - Complex Form of Fourier Series.

Unit-V

Partial Differential Equations & Applications : Formation of Partial Differential Equations by Elimination of Constants and Arbitrary Functions - Solution of First Order PDEs by Lagrange’s Method - Solution of non Linear PDEs (Standard Forms) - Solution of Second Order PDEs by Method of Separation of Variables - Solutions of One Dimensional Wave Equation, One Dimensional Heat Equation under Initial and Boundary Conditions