# Complex Variables and Numerical Methods (2141905)

Syllabus

Sr. Topics Teaching Hours Module Weightage
1
Complex Numbers and Functions:

Exponential, Trigonometric, De Moivre’s Theorem, Roots of a complex number ,Hyperbolic functions and their properties, Multi-valued function and its branches: Logarithmic function and Complex Exponent function Limit ,Continuity and Differentiability of complex function, Analytic functions, Cauchy-Riemann Equations, Necessary and Sufficient condition for analyticity, Properties of Analytic functions, Laplace equation, Harmonic Functions, Harmonic Conjugate functions and their Engineering Applications

10
24 %
2
Complex Integration:

Curves, Line Integral(contour integral) and its properties, Cauchy-Goursat Theorem, Cauchy Integral Formula, Liouville Theorem (without proof), Maximum Modulus Theorems(without proof)

4
10 %
3
Power Series:

Convergence(Ordinary, Uniform, Absolute) of power series, Taylor and Laurent Theorems (without proof), Laurent series expansions, zeros of analytic functions , Singularities of analytic functions and their classification Residues: Residue Theorem, Rouche’s Theorem (without proof)

5
12 %
4
Applications of Contour Integration:

Evaluation of various types of definite real integrals using contour integration method

2
5 %
5
Conformal Mapping and its Applications:

Conformal and Isogonal mappings , Translation, Rotation & Magnification, Inversion, Mobius(Bilinear) , Schwarz-Christoffel transformations

3
7 %
6
Interpolation: Finite Differences, Forward, Backward and Central operators, Interpolation by polynomials:

Newton’s forward ,Backward interpolation formulae, Newton’s divided Gauss & Stirling’s central difference formulae and Lagrange’s interpolation formulae for unequal intervals

4
10 %
7
Numerical Integration:

Newton-Cotes formula, Trapezoidal and Simpson’s formulae, error formulae, Gaussian quadrature formulae

3
7 %
8
Solution of a System of Linear Equations:

Gauss elimination, partial pivoting , Gauss-Jacobi method and Gauss-Seidel method

3
7 %
9
Roots of Algebraic and Transcendental Equations :

Bisection, false position, Secant and Newton-Raphson methods, Rate of convergence

3
7 %
10
Eigen values by Power and Jacobi methods
2
4 %
11
Numerical solution of Ordinary Differential Equations:

Euler and Runge-Kutta methods

3
7 %