## Syllabus

# UNIT-I

**Functions of Complex Variables**: Limits and continuity of function, differentiability and analyticity, necessary & sufficient conditions for a function to be analytic, Cauchy- Reimann equations in polar form, harmonic functions, complex integration, Cauchy’s integral theorem, extension of Cauchy’s integral theorem for multiply connected regions, Cauchy’s integral formula, Cauchy’s formula for derivatives and their applications.

# UNIT-II

**Residue Calculus: **Power series, Taylor’s series, Laurent’s series, zeros and singularities, residues, residue theorem, evaluation of real integrals using residue theorem, bilinear transformation, and conformal mapping.

# UNIT-III

**Fourier Series: **Fourier series, Fourier series expansions of even and odd functions, convergence of Fourier series, and Fourier half range series.

# UNIT-IV

**Partial Differential Equations**: Formation of first and second order partial differential equations, solution of first order equations, Lagrange’s equation, Nonlinear first order equations, Charpit’s method, higher order linear equations with constant coefficients.

# UNIT-V

**Fourier Series Applications to Partial Differential Equations: **Classification of linear second order partial differential equations, Separation of variables method (Fourier method), Fourier series solution of one dimensional heat and wave equations, Laplace’s equation.