Explore advanced mathematical concepts in this comprehensive 34-hour course covering complex analysis, Laplace and Fourier transforms, probability theory, linear algebra, and more. Delve into topics like analytic functions, Cauchy integral theorem, power series, singularities, and residue theorem. Learn about Laplace and Fourier transforms and their applications to partial differential equations. Study probability laws, random variables, and special distributions. Investigate vector spaces, matrices, eigenvalues, and linear transformations. Master orthogonality, inner product spaces, and the Gram-Schmidt process. Gain a deep understanding of advanced engineering mathematics to enhance your problem-solving skills in various scientific and engineering fields.
Overview
Syllabus
Analytic Functions, C-R Equations.
Harmonic Functions.
Line Integral in the Complex.
Cauchy Integral Theorem.
Cauchy Integral Theorem (Contd.).
Cauchy Integral Formula.
Power and Taylor Series of Complex Numbers.
Power and Taylor Series of Complex Numbers (Contd.).
Taylor's , Laurent Series of f(z) and Singularities.
Classification of Singularities, Residue and Residue Theorem.
Laplace Transform and its Existence.
Properties of Laplace Transform.
Evaluation of Laplace and Inverse Laplace Transform.
S30 2072.
Applications of Laplace Transform to PDEs.
Fourier Series (Contd.).
Fourier Integral Representation of a Function.
Introduction to Fourier Transform.
Applications of Fourier Transform to PDEs.
Laws of probability I.
Laws of probability II.
Problems in probability.
Random variables.
Special Discrete Distributions.
Special Continuous distributions.
Vector Spaces, Subspaces, Linearly Dependent / Independent of Vectors.
Review Groups, Fields and Matrices.
Basis, Dimension, Rank and Matrix Inverse.
Jordan Canonical Form,Cayley Hamilton Theorem.
Concept of Domain, Limit, Continuity and Differentiability.
Method to Find Eigenvalues and Eigenvectors, Diagonalization of Matrices.
Spectrum of special matrices,positive/negative definite matrices.
System of Linear Equations, Eigen values and Eigen vectors.
Linear Transformation, Isomorphism and Matrix Representation.
Orthogonality , Gram-Schmidt Orthogonalization Process.
Inner Product Spaces, Cauchy - Schwarz Inequality.
Taught by
Ch 30 NIOS: Gyanamrit
