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
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