# Mathematics-4 (140001)

Syllabus

Sr. Topics Teaching Hours Module Weightage
1
Complex numbers and functions
Limits of Functions, Continuity, Differentiability, Analytic functions, Cauchy-Riemann Equations, Necessary and Sufficient condition for analyticity, Properties of Analytic Functions, Laplace Equation, Harmonic Functions, Finding Harmonic Conjugate functions Exponential, Trigonometric, Hyperbolic functions and its properties. Multiple valued function and its branches: Logarithmic function and Complex Exponent function
2
Complex Integration
Curves, Line Integrals (contour integral) and its properties. Line integrals of single valued functions, Line integrals of multiple valued functions (by choosing suitable branches). Cauchy-Goursat Theorem, Cauchy Integral Formula, Liouville Theorem, Fundamental Theorem of Algebra, Maximum Modulus Theorems
3
Power Series
Convergence (Ordinary, Uniform, Absolute) of power series, Taylor and Laurent Theorems, Laurent series expansions. Zeros of analytic functions. Singularities of analytic functions and their classification Residues: Residue Theorem, Rouche’s Theorem, Argument Principle
4
Applications of Contour Integration
Evaluating various type of definite real integrals using contour integration method
5
Conformal Mapping and its applications
Mappings by elementary functions, Mobius transformations, Schwarz- Christoffel transformation
6
Interpolation
Interpolation by polynomials, divided differences, error of the interpolating polynomial
7
Numerical integration
Composite rules, error formulae, Gaussian integration
8
Linear algebraic equation
Solution of a system of linear equations: implementation of Gaussian elimination and Gauss-Seidel methods, partial pivoting
9
Roots of equation
Solution of a nonlinear equation: Bisection and Secant methods, Newton’s method, rate of convergence, Power method for computation of Eigen values
10
Ordinary differential equations
Numerical solution of ordinary differential equations, Euler and Runge- Kutta methods