Differential Equations - Theory of Function of Complex Variable

  • Introduction to differential equations
  • Definition of a complex variable
  • Introduction to theory of functions of a complex variable
  • Why study differential equations in complex variables?
  • Applications of differential equations in complex variables

Complex Numbers

  • Definition of a complex number
  • Real and imaginary parts of a complex number
  • Basic arithmetic operations on complex numbers
  • Polar form of a complex number
  • Euler’s formula

Analytic Functions

  • Definition of an analytic function
  • Differentiability of an analytic function
  • Cauchy-Riemann equations
  • Harmonic functions

Line Integrals

  • Definition of a line integral
  • Path independence of a line integral
  • Cauchy’s theorem for an analytic function
  • Proof of Cauchy’s theorem using Green’s theorem
  • Contour integrals

Cauchy’s Integral Formula

  • Statement of Cauchy’s integral formula
  • Calculation of contour integrals using Cauchy’s integral formula
  • Examples of contour integrals using Cauchy’s integral formula
  • Singularities of an analytic function
  • Residue theorem

Laurent Series and Residues

  • Expansion of an analytic function using Laurent series
  • Definition of a residue
  • Calculation of residues
  • Residue theorem and its applications
  • Evaluation of real integrals using residues

Conformal Mapping

  • Definition of conformal mapping
  • Examples of conformal mapping
  • Mapping of circles and straight lines using conformal mapping
  • Applications of conformal mapping in fluid dynamics

Singularities

  • Classification of singularities
  • Essential singularities
  • Removable singularities
  • Poles
  • Infinite singularities

Complex Integration Methods

  • Method of residues
  • Partial fractions decomposition
  • Cauchy’s residue theorem
  • Residue at infinity
  • Jordan’s lemma

Applications of Complex Variables

  • Engineering applications
  • Physics applications
  • Signal processing applications
  • Image processing applications
  • Control systems applications

Differential Equations - Theory of Function of Complex Variable

  • Introduction to differential equations
  • Definition of a complex variable
  • Introduction to theory of functions of a complex variable
  • Why study differential equations in complex variables?
  • Applications of differential equations in complex variables

Complex Numbers

  • Definition of a complex number
  • Real and imaginary parts of a complex number
  • Basic arithmetic operations on complex numbers
  • Polar form of a complex number
  • Euler’s formula

Analytic Functions

  • Definition of an analytic function
  • Differentiability of an analytic function
  • Cauchy-Riemann equations
  • Harmonic functions
  • Examples of analytic functions

Line Integrals

  • Definition of a line integral
  • Path independence of a line integral
  • Cauchy’s theorem for an analytic function
  • Proof of Cauchy’s theorem using Green’s theorem
  • Contour integrals

Cauchy’s Integral Formula

  • Statement of Cauchy’s integral formula
  • Calculation of contour integrals using Cauchy’s integral formula
  • Examples of contour integrals using Cauchy’s integral formula
  • Singularities of an analytic function
  • Residue theorem

Laurent Series and Residues

  • Expansion of an analytic function using Laurent series
  • Definition of a residue
  • Calculation of residues using Laurent series
  • Residue theorem and its applications
  • Evaluation of real integrals using residues

Conformal Mapping

  • Definition of conformal mapping
  • Examples of conformal mapping
  • Mapping of circles and straight lines using conformal mapping
  • Applications of conformal mapping in fluid dynamics
  • Properties of conformal mapping

Singularities

  • Classification of singularities
  • Essential singularities
  • Removable singularities
  • Poles
  • Infinite singularities

Complex Integration Methods

  • Method of residues
  • Partial fractions decomposition
  • Cauchy’s residue theorem
  • Residue at infinity
  • Jordan’s lemma

Applications of Complex Variables

  • Engineering applications
  • Physics applications
  • Signal processing applications
  • Image processing applications
  • Control systems applications

Differential Equations - Theory of Function of Complex Variable

  • Definition and classification of differential equations
  • Order and degree of a differential equation
  • Solutions of a differential equation
  • General solution and particular solution of a differential equation
  • Examples of differential equations

Linear Differential Equations

  • Definition of a linear differential equation
  • Homogeneous and non-homogeneous linear differential equations
  • Solution of a homogeneous linear differential equation
  • Solution of a non-homogeneous linear differential equation
  • Examples of linear differential equations

Laplace Transforms

  • Definition of Laplace transform
  • Laplace transform of standard functions
  • Properties of Laplace transforms
  • Inverse Laplace transform
  • Applications of Laplace transforms in solving differential equations

Fourier Series

  • Definition of Fourier series
  • Coefficients of Fourier series
  • Even and odd functions
  • Half-range Fourier series
  • Applications of Fourier series in solving differential equations

Partial Differential Equations

  • Introduction to partial differential equations
  • Classification of partial differential equations
  • Solution methods for partial differential equations
  • Examples of partial differential equations
  • Applications of partial differential equations in physics and engineering

Boundary Value Problems

  • Definition and examples of boundary value problems
  • Solution methods for boundary value problems
  • Green’s functions and their applications in boundary value problems
  • Examples of boundary value problems
  • Applications of boundary value problems in physics and engineering

Numerical Methods for Differential Equations

  • Introduction to numerical methods for differential equations
  • Euler’s method and its limitations
  • Runge-Kutta methods
  • Finite Difference methods
  • Applications of numerical methods in solving differential equations

Systems of Differential Equations

  • Definition of a system of differential equations
  • Solution methods for systems of differential equations
  • Eigenvalues and eigenvectors of a matrix
  • Stability of solutions of a system of differential equations
  • Applications of systems of differential equations in physics and engineering

Nonlinear Differential Equations

  • Introduction to nonlinear differential equations
  • Solution methods for nonlinear differential equations
  • Phase portraits and stability analysis for nonlinear systems
  • Chaos and chaotic behavior in nonlinear systems
  • Applications of nonlinear differential equations in physics and engineering

Applications of Differential Equations

  • Engineering applications of differential equations
  • Physics applications of differential equations
  • Biological applications of differential equations
  • Economics applications of differential equations
  • Environmental applications of differential equations