Special functions: Gamma and Beta Functions – Their properties – evaluation of improper integrals. Bessel functions – properties – Recurrence relations – Orthogonality. Legendre polynomials – Properties – Rodrigue’s formula – Recurrence relations – Orthogonality.

UNIT-II

Functions of a complex variable – Continuity – Differentiability – Analyticity – Properties – Cauchy-Riemann equations in Cartesian and polar coordinates. Harmonic and conjugate harmonic functions – Milne – Thompson method.

UNIT-III

Elementary functions: Exponential, trigonometric, hyperbolic functions and their properties – General power Z  (c is complex), principal value.

UNIT-IV

Complex integration: Line integral – evaluation along a path and by indefinite integration – Cauchy’s integral theorem – Cauchy’s integral formula – Generalized integral formula.

UNIT-V

Complex power series: Radius of convergence – Expansion in Taylor’s series, Maclaurin’s series and Laurent series. Singular point –Isolated singular point – pole of order m – essential singularity.

UNIT-VI

Residue – Evaluation of residue by formula and by Laurent series - Residue theorem.

Evaluation of integrals of the type

(a) Improper real integrals              (b)    

(c)                                    (d) Integrals by identation.

UNIT-VII

Argument principle – Rouche’s theorem – determination of number of zeros of complex polynomials - Maximum Modulus principle - Fundamental theorem of Algebra, Liouville’s Theorem.

UNIT-VIII

Conformal mapping: Transformation by , lnz, z², z (n positive integer), Sin z, cos z,                z + a/z. Translation, rotation, inversion and bilinear transformation – fixed point – cross ratio – properties – invariance of circles and cross ratio – determination of bilinear transformation mapping 3 given points .

Chapter One - Complex Numbers
  1.1 Introduction
  1.2 Geometry
  1.3 Polar coordinates

Chapter Two - Complex Functions
  2.1 Functions of a real variable
  2.2 Functions of a complex variable
  2.3 Derivatives

Chapter Three - Elementary Functions
  3.1 Introduction
  3.2 The exponential function
  3.3 Trigonometric functions
  3.4 Logarithms and complex exponents

Chapter Four - Integration
  4.1 Introduction
  4.2 Evaluating integrals
  4.3 Antiderivatives

Chapter Five - Cauchy's Theorem
  5.1 Homotopy
  5.2 Cauchy's Theorem

Chapter Six - More Integration
  6.1 Cauchy's Integral Formula
  6.2 Functions defined by integrals
  6.3 Liouville's Theorem
  6.4 Maximum moduli

Chapter Seven - Harmonic Functions
  7.1 The Laplace equation
  7.2 Harmonic functions
  7.3 Poisson's integral formula

Chapter Eight - Series
  8.1 Sequences
  8.2 Series
  8.3 Power series
  8.4 Integration of power series
  8.5 Differentiation of power series

Chapter Nine - Taylor and Laurent Series
  9.1 Taylor series
  9.2 Laurent series

Chapter Ten - Poles, Residues, and All That
  10.1 Residues
  10.2 Poles and other singularities