Lecture

Mod-02 Lec-12 Analytic Functions, C-R Equations

This module covers analytic functions and Cauchy-Riemann equations. Students will learn to:

  • Define analytic functions and their properties
  • Apply Cauchy-Riemann equations for determining analyticity
  • Explore applications of analytic functions in engineering

Understanding these concepts is essential for advanced studies in complex analysis.


Course Lectures
  • Mod-01 Lec-01 Review Groups, Fields and Matrices
    Dr. P. Panigrahi, Prof. J. Kumar, Prof. P.D. Srivastava, Prof. Somesh Kumar

    This module introduces the fundamental concepts of groups, fields, and matrices. Understanding these concepts is crucial for further studies in linear algebra. Students will learn the definitions and properties of:

    • Groups
    • Fields
    • Matricial operations

    Real-world applications will also be discussed, emphasizing the importance of these topics in various engineering fields.

  • Mod-01 Lec-02 Vector Spaces, Subspaces, Linearly Dependent/Independent of Vectors
    Dr. P. Panigrahi, Prof. J. Kumar, Prof. P.D. Srivastava, Prof. Somesh Kumar

    This module focuses on vector spaces and their critical components, including subspaces, linearly dependent and independent vectors. Students will learn how to:

    • Identify vector spaces and their properties
    • Determine the linear dependence of given vectors
    • Understand concepts of basis and dimension

    These concepts are foundational for advanced studies in mathematics and engineering.

  • Mod-01 Lec-03 Basis, Dimension, Rank and Matrix Inverse
    Dr. P. Panigrahi, Prof. J. Kumar, Prof. P.D. Srivastava, Prof. Somesh Kumar

    This module delves into the concepts of basis, dimension, rank, and matrix inverse. These topics are essential for understanding the behavior of linear transformations. Students will learn to:

    • Calculate the rank of a matrix
    • Find the inverse of a matrix when it exists
    • Apply these concepts in solving linear equations

    Real-world applications of these mathematical tools will also be emphasized, ensuring students grasp their significance.

  • Mod-01 Lec-04 Linear Transformation, Isomorphism and Matrix Representation
    Dr. P. Panigrahi, Prof. J. Kumar, Prof. P.D. Srivastava, Prof. Somesh Kumar

    This module introduces linear transformations and their matrix representations. Students will learn how to:

    • Define linear transformations
    • Understand isomorphism and its implications
    • Represent transformations using matrices

    By the end of this module, students will have a solid understanding of how linear transformations operate in vector spaces.

  • Mod-01 Lec-05 System of Linear Equations, Eigenvalues and Eigenvectors
    Dr. P. Panigrahi, Prof. J. Kumar, Prof. P.D. Srivastava, Prof. Somesh Kumar

    This module focuses on systems of linear equations, eigenvalues, and eigenvectors. Students will learn to:

    • Set up and solve systems of linear equations
    • Calculate eigenvalues and eigenvectors
    • Understand the significance of these concepts in applications

    The module will also cover the geometric interpretation of eigenvalues and eigenvectors, enhancing students' comprehension.

  • Mod-01 Lec-06 Method to Find Eigenvalues and Eigenvectors, Diagonalization of Matrices
    Dr. P. Panigrahi, Prof. J. Kumar, Prof. P.D. Srivastava, Prof. Somesh Kumar

    This module discusses methods for finding eigenvalues and eigenvectors, as well as the diagonalization of matrices. Students will learn:

    • Techniques to calculate eigenvalues and eigenvectors
    • How to diagonalize a matrix when possible
    • The implications of diagonalization in various applications

    This foundational knowledge will support students in future studies involving linear transformations.

  • Mod-01 Lec-07 Jordan Canonical Form, Cayley Hamilton Theorem
    Dr. P. Panigrahi, Prof. J. Kumar, Prof. P.D. Srivastava, Prof. Somesh Kumar

    This module covers the Jordan canonical form and the Cayley-Hamilton theorem. Students will learn to:

    • Define and compute the Jordan form of a matrix
    • Understand the Cayley-Hamilton theorem and its applications
    • Explore the significance of these concepts in engineering

    By mastering these topics, students will develop a deeper understanding of matrix theory and its applications.

  • Mod-01 Lec-08 Inner Product Spaces, Cauchy-Schwarz Inequality
    Dr. P. Panigrahi, Prof. J. Kumar, Prof. P.D. Srivastava, Prof. Somesh Kumar

    This module introduces inner product spaces and the Cauchy-Schwarz inequality. Students will learn to:

    • Define inner product spaces and their properties
    • Apply the Cauchy-Schwarz inequality in various contexts
    • Understand the implications of these concepts in engineering

    These foundational concepts will provide a basis for further studies in functional analysis and other advanced mathematics topics.

  • Mod-01 Lec-09 Orthogonality, Gram-Schmidt Orthogonalization Process
    Dr. P. Panigrahi, Prof. J. Kumar, Prof. P.D. Srivastava, Prof. Somesh Kumar

    This module focuses on orthogonality and the Gram-Schmidt orthogonalization process. Students will learn how to:

    • Define orthogonality in vector spaces
    • Employ the Gram-Schmidt process to create orthogonal sets
    • Apply these concepts in various engineering contexts

    Understanding these principles is crucial for advanced studies in linear algebra and related fields.

  • Mod-01 Lec-10 Spectrum of special matrices,positive/negative definite matrices
    Dr. P. Panigrahi, Prof. J. Kumar, Prof. P.D. Srivastava, Prof. Somesh Kumar

    This module explores the spectrum of special matrices, including positive and negative definite matrices. Students will learn to:

    • Identify special matrices and their properties
    • Determine positive and negative definiteness
    • Apply these concepts in engineering applications

    Understanding these matrix properties is essential for many advanced mathematical topics.

  • Mod-02 Lec-11 Concept of Domain, Limit, Continuity and Differentiability
    Dr. P. Panigrahi, Prof. J. Kumar, Prof. P.D. Srivastava, Prof. Somesh Kumar

    This module introduces the concepts of domain, limit, continuity, and differentiability in calculus. Students will learn how to:

    • Define and analyze function limits
    • Understand continuity and its implications
    • Explore differentiability and its applications

    These foundational concepts are crucial for understanding advanced calculus and its applications in engineering.

  • Mod-02 Lec-12 Analytic Functions, C-R Equations
    Dr. P. Panigrahi, Prof. J. Kumar, Prof. P.D. Srivastava, Prof. Somesh Kumar

    This module covers analytic functions and Cauchy-Riemann equations. Students will learn to:

    • Define analytic functions and their properties
    • Apply Cauchy-Riemann equations for determining analyticity
    • Explore applications of analytic functions in engineering

    Understanding these concepts is essential for advanced studies in complex analysis.

  • Mod-02 Lec-13 Harmonic Functions
    Dr. P. Panigrahi, Prof. J. Kumar, Prof. P.D. Srivastava, Prof. Somesh Kumar

    This module introduces harmonic functions and their significance in engineering. Students will learn:

    • Define harmonic functions and their properties
    • Explore the Laplace equation
    • Apply harmonic functions in various engineering scenarios

    Mastering these concepts is crucial for further studies in partial differential equations and applied mathematics.

  • Mod-02 Lec-14 Line Integral in the Complex
    Dr. P. Panigrahi, Prof. J. Kumar, Prof. P.D. Srivastava, Prof. Somesh Kumar

    This module discusses line integrals in the complex plane. Students will learn to:

    • Define line integrals and their significance
    • Calculate line integrals for complex functions
    • Apply these integrals in engineering applications

    Understanding line integrals is essential for advanced studies in complex analysis and engineering mathematics.

  • Mod-02 Lec-15 Cauchy Integral Theorem
    Dr. P. Panigrahi, Prof. J. Kumar, Prof. P.D. Srivastava, Prof. Somesh Kumar

    This module focuses on the Cauchy Integral Theorem, a fundamental concept in complex analysis. Students will learn:

    • The statement and proof of the theorem
    • Applications of the theorem in evaluating integrals
    • Implications for analytic functions

    Mastering this theorem is crucial for advanced studies in complex analysis.

  • Mod-02 Lec-16 Cauchy Integral Theorem (Contd.)
    Dr. P. Panigrahi, Prof. J. Kumar, Prof. P.D. Srivastava, Prof. Somesh Kumar

    This module continues the exploration of the Cauchy Integral Theorem, further delving into its applications in complex analysis. Students will learn how to:

    • Apply the theorem to complex integrals
    • Analyze the implications for contour integration
    • Explore advanced applications in engineering

    Understanding these concepts will enhance students’ knowledge of complex function theory.

  • Mod-02 Lec-17 Cauchy Integral Formula
    Dr. P. Panigrahi, Prof. J. Kumar, Prof. P.D. Srivastava, Prof. Somesh Kumar

    This module introduces the Cauchy Integral Formula, building on the Cauchy Integral Theorem. Students will learn:

    • The statement and proof of the formula
    • Applications of the formula in evaluating complex integrals
    • Implications for analytic functions

    Mastering this formula is essential for advanced studies in complex analysis and its applications in engineering.

  • Mod-02 Lec-18 Power and Taylor's Series of Complex Numbers
    Dr. P. Panigrahi, Prof. J. Kumar, Prof. P.D. Srivastava, Prof. Somesh Kumar

    This module discusses power and Taylor series of complex numbers. Students will learn to:

    • Define and derive power series
    • Understand the convergence of Taylor series
    • Apply these series in engineering problems

    These concepts are foundational for further studies in series and functions in complex analysis.

  • Mod-02 Lec-19 Power and Taylor's Series of Complex Numbers (Contd.)
    Dr. P. Panigrahi, Prof. J. Kumar, Prof. P.D. Srivastava, Prof. Somesh Kumar

    This module continues the study of power and Taylor series of complex numbers, providing further insights into their properties and applications. Students will learn:

    • Advanced techniques for series expansion
    • Applications in engineering problems
    • Connections with analytic functions

    These advanced concepts are crucial for mastering complex analysis.

  • Mod-02 Lec-20 Taylor's, Laurent Series of f(z) and Singularities
    Dr. P. Panigrahi, Prof. J. Kumar, Prof. P.D. Srivastava, Prof. Somesh Kumar

    This module covers Taylor's and Laurent series for complex functions and discusses singularities. Students will learn:

    • The definitions and properties of Taylor and Laurent series
    • How to identify singularities of complex functions
    • Applications in engineering contexts

    Understanding these series is crucial for advanced studies in complex analysis and its applications.

  • Mod-02 Lec-21 Classification of Singularities, Residue and Residue Theorem
    Dr. P. Panigrahi, Prof. J. Kumar, Prof. P.D. Srivastava, Prof. Somesh Kumar

    This module discusses the classification of singularities and the residue theorem. Students will learn to:

    • Classify singularities of complex functions
    • Apply the residue theorem in complex integrals
    • Explore the implications for engineering applications

    Mastering these concepts is crucial for advanced studies in complex analysis and its practical applications in engineering.

  • Mod-03 Lec-22 Laplace Transform and its Existence
    Dr. P. Panigrahi, Prof. J. Kumar, Prof. P.D. Srivastava, Prof. Somesh Kumar

    This module introduces the Laplace transform and its existence. Students will learn:

    • The definition and application of the Laplace transform
    • Conditions for the existence of the transform
    • Real-world applications in engineering

    Understanding the Laplace transform is crucial for solving differential equations in engineering contexts.

  • Mod-03 Lec-23 Properties of Laplace Transform
    Dr. P. Panigrahi, Prof. J. Kumar, Prof. P.D. Srivastava, Prof. Somesh Kumar

    This module explores the properties of the Laplace transform, including linearity and shifting. Students will learn to:

    • Analyze the properties of the Laplace transform
    • Apply these properties in engineering problems
    • Understand implications for differential equations

    Mastering these properties is essential for advanced studies in transform calculus.

  • Mod-03 Lec-25 Applications of Laplace Transform to Integral Equations and ODEs
    Dr. P. Panigrahi, Prof. J. Kumar, Prof. P.D. Srivastava, Prof. Somesh Kumar

    This module discusses the applications of the Laplace transform to integral equations and ordinary differential equations (ODEs). Students will learn:

    • How to apply the Laplace transform to solve ODEs
    • Applications in engineering problems
    • Real-world scenarios where Laplace transforms are beneficial

    Mastering these applications is crucial for advanced engineering mathematics.

  • Mod-03 Lec-26 Applications of Laplace Transform to PDEs
    Dr. P. Panigrahi, Prof. J. Kumar, Prof. P.D. Srivastava, Prof. Somesh Kumar

    This module discusses the applications of the Laplace transform to partial differential equations (PDEs). Students will learn:

    • How to apply the Laplace transform to solve PDEs
    • Real-world applications in engineering
    • Techniques for transforming complex problems

    Mastering these applications is essential for advanced studies in engineering and applied mathematics.

  • Mod-03 Lec-27 Fourier Series
    Dr. P. Panigrahi, Prof. J. Kumar, Prof. P.D. Srivastava, Prof. Somesh Kumar

    This module introduces Fourier series, discussing their definitions, convergence, and applications. Students will learn:

    • The concept of Fourier series and their significance
    • How to determine convergence of Fourier series
    • Applications in engineering problems

    Understanding Fourier series is crucial for advanced studies in signal processing and related fields.

  • Mod-03 Lec-28 Fourier Series (Contd.)
    Dr. P. Panigrahi, Prof. J. Kumar, Prof. P.D. Srivastava, Prof. Somesh Kumar

    This module continues the discussion on Fourier series, focusing on their applications in various engineering contexts. Students will learn:

    • Advanced applications of Fourier series
    • Real-world scenarios where Fourier series are beneficial
    • Connections to other mathematical concepts

    Mastering these applications is crucial for advanced engineering mathematics and signal processing.

  • Mod-03 Lec-29 Fourier Integral Representation of a Function
    Dr. P. Panigrahi, Prof. J. Kumar, Prof. P.D. Srivastava, Prof. Somesh Kumar

    This module introduces Fourier integral representation of a function. Students will learn:

    • The concept and significance of Fourier integrals
    • How to represent functions using Fourier integrals
    • Applications in engineering problems

    Understanding Fourier integrals is essential for advanced studies in signal analysis and other fields.

  • Mod-03 Lec-30 Introduction to Fourier Transform
    Dr. P. Panigrahi, Prof. J. Kumar, Prof. P.D. Srivastava, Prof. Somesh Kumar

    This module introduces the Fourier transform and its applications in engineering. Students will learn:

    • The definition and significance of the Fourier transform
    • How to apply the Fourier transform to engineering problems
    • Real-world applications in signal processing

    Mastering the Fourier transform is crucial for advanced studies in engineering mathematics and applied fields.

  • Mod-03 Lec-31 Applications of Fourier Transform to PDEs
    Dr. P. Panigrahi, Prof. J. Kumar, Prof. P.D. Srivastava, Prof. Somesh Kumar

    This module discusses the applications of the Fourier transform to partial differential equations (PDEs). Students will learn:

    • How to apply the Fourier transform to solve PDEs
    • Real-world applications in engineering
    • Techniques for transforming complex problems

    Mastering these applications is essential for advanced studies in engineering and applied mathematics.

  • Mod-04 Lec-32 Laws of Probability - I
    Dr. P. Panigrahi, Prof. J. Kumar, Prof. P.D. Srivastava, Prof. Somesh Kumar

    This module covers the laws of probability, introducing basic concepts and principles. Students will learn:

    • The fundamental laws of probability
    • How to apply these laws in various contexts
    • Real-world applications in engineering and statistics

    Understanding probability laws is essential for advanced studies in statistics and data analysis.

  • Mod-04 Lec-33 Laws of Probability - II
    Dr. P. Panigrahi, Prof. J. Kumar, Prof. P.D. Srivastava, Prof. Somesh Kumar

    This module continues the discussion on probability laws, focusing on advanced concepts and their applications. Students will learn:

    • Advanced probability laws and their significance
    • Applications in engineering and real-world scenarios
    • Connections to other mathematical concepts

    Mastering these concepts is crucial for advanced studies in probability and statistics.

  • Mod-04 Lec-34 Problems in Probability
    Dr. P. Panigrahi, Prof. J. Kumar, Prof. P.D. Srivastava, Prof. Somesh Kumar

    This module discusses various problems in probability, providing students with practical applications. Students will learn:

    • How to approach and solve probability problems
    • Applications in engineering and real-world scenarios
    • Problem-solving techniques and strategies

    Understanding these concepts is essential for advanced studies in statistics and data analysis.

  • Mod-04 Lec-35 Random Variables
    Dr. P. Panigrahi, Prof. J. Kumar, Prof. P.D. Srivastava, Prof. Somesh Kumar

    This module introduces random variables and their significance in probability theory. Students will learn:

    • The definition and types of random variables
    • How to calculate expected values and variances
    • Applications in engineering and statistics

    Understanding random variables is crucial for advanced studies in probability and statistics.

  • Mod-04 Lec-36 Special Discrete Distributions
    Dr. P. Panigrahi, Prof. J. Kumar, Prof. P.D. Srivastava, Prof. Somesh Kumar

    This module discusses special discrete distributions and their applications in probability. Students will learn:

    • The characteristics of special discrete distributions
    • Applications in engineering and statistics
    • How to use these distributions in real-world scenarios

    Understanding these distributions is essential for advanced studies in probability and statistics.

  • Mod-04 Lec-37 Special Continuous Distributions
    Dr. P. Panigrahi, Prof. J. Kumar, Prof. P.D. Srivastava, Prof. Somesh Kumar

    This module covers special continuous distributions and their significance in probability theory. Students will learn:

    • The characteristics of special continuous distributions
    • Applications in engineering and statistics
    • How to utilize these distributions in real-world problems

    Understanding these distributions is crucial for advanced studies in probability and statistics.

  • Mod-04 Lec-38 Joint Distributions and Sampling Distributions
    Dr. P. Panigrahi, Prof. J. Kumar, Prof. P.D. Srivastava, Prof. Somesh Kumar

    This module introduces joint distributions and sampling distributions, discussing their significance in probability. Students will learn:

    • The definitions and properties of joint distributions
    • How to analyze sampling distributions
    • Applications in engineering and statistics

    Mastering these concepts is essential for advanced studies in probability and statistics.

  • Mod-04 Lec-39 Point Estimation
    Dr. P. Panigrahi, Prof. J. Kumar, Prof. P.D. Srivastava, Prof. Somesh Kumar

    This module covers point estimation in statistics, discussing its importance in data analysis. Students will learn:

    • The concepts of point estimation and its techniques
    • Applications in engineering and real-world scenarios
    • How to calculate point estimates from data

    Understanding point estimation is crucial for advanced studies in statistics and data analysis.

  • Mod-04 Lec-40 Interval Estimation
    Dr. P. Panigrahi, Prof. J. Kumar, Prof. P.D. Srivastava, Prof. Somesh Kumar

    This module discusses interval estimation in statistics, providing insights into its significance. Students will learn:

    • The concept of interval estimation and its techniques
    • Applications in engineering and real-world scenarios
    • How to calculate confidence intervals from data

    Mastering interval estimation techniques is essential for advanced studies in statistics and data analysis.

  • Mod-04 Lec-41 Basic Concepts of Testing of Hypothesis
    Dr. P. Panigrahi, Prof. J. Kumar, Prof. P.D. Srivastava, Prof. Somesh Kumar

    This module introduces the basic concepts of hypothesis testing in statistics. Students will learn:

    • The fundamental principles of hypothesis testing
    • How to conduct tests and interpret results
    • Applications in engineering and real-world scenarios

    Understanding hypothesis testing is crucial for advanced studies in statistics and data analysis.

  • Mod-04 Lec-42 Tests for Normal Populations
    Dr. P. Panigrahi, Prof. J. Kumar, Prof. P.D. Srivastava, Prof. Somesh Kumar

    This module covers tests for normal populations, discussing their significance in statistics. Students will learn:

    • The principles and techniques for testing normal populations
    • Applications in engineering and real-world scenarios
    • How to conduct tests and interpret results

    Mastering these concepts is essential for advanced studies in statistics and data analysis.