Contents

• Unit-1 Introduction: Objective, scope and outcome of the course.

   

  Unit-2 Complex Variable – Differentiation: Differentiation, Cauchy-Riemann equations, analytic functions, harmonic functions, finding harmonic conjugate; elementary analytic functions (exponential, trigonometric, logarithm) and their properties; Conformal mappings, Mobius transformations and their properties.

   

  Unit-3 Complex Variable - Integration: Contour integrals, Cauchy-Goursat theorem (without proof), Cauchy Integral formula (without proof), Liouville’s theorem and Maximum- Modulus theorem (without proof); Taylor’s series, zeros of analytic functions, singularities, Laurent’s series; Residues, Cauchy Residue theorem (without proof).

   

  Unit-4 Applications of complex integration by residues: Evaluation of definite integral involving sine and cosine. Evaluation of certain improper integrals.

   

  Unit-5 Special Functions: Legendre’s function, Rodrigues formula, generating function, Simple recurrence relations, orthogonal property. Bessel’s functions of first and second kind, generating function, simple recurrence relations, orthogonal property.

   

  Unit-6 Linear Algebra: Vector Spaces, subspaces, Linear independence, basis and dimension, Inner product spaces, Orthogonality, Gram Schmidt orthogonalization, characteristic polynomial, minimal polynomial, positive definite matrices and canonical forms, QR decomposition.