Learn · Mathematical Methods In Physics 2
Lecture 8 of 80Week 1
Cauchy-Riemann Equations
Prof. Auditya Sharma · IISER Bhopal
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- Mathematical Methods In Physics 2, IISER Bhopal (at NPTEL)
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- CC BY-NC-SA, as NPTEL states it. Non-commercial use only.
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- NPTEL states: "Distributed under Creative Commons Attribution-ShareAlike / CC BY - NC - SA", beside a BY-NC-SA badge. Coordinated by IIT Madras.
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All the lectures
- Introduction to complex numbers
- The triangle inequality
- The de Moivre formula
- Roots of unity
- Functions of a complex variable and the notion of continuity
- Derivative of a complex function
- Differentiation rules for a complex function
- Cauchy-Riemann Equations
- Sufficient conditions for differentiability
- Cauchy-Riemann conditions in polar coordinates
- More persepective on differentiability
- The value of the derivative
- Analytic functions
- Harmonic functions
- The exponential function
- Complex logarithm
- Complex exponents
- Trigonometric functions of complex variables
- Hyperbolic functions of complex variables
- Inverse Trigonometric and Hyperbolic functions
- Branch of a multivalued function
- Contour Integrals
- Green's Theorem
- Path dependence of the contour intergal
- Antiderivatives
- The Cauchy theorem
- Crossing contours and multiply connected domains
- Cauchy Integral formula
- Derivatives of an analytic function
- Liouville's theorem and the Fundamental theorem of algebra
- Taylor Series
- Laurent Series
- Convergence
- Differentiation and integration of power series
- Isolated Singularities
- Residues
- Residue Theorem
- Evaluation of integrals-I
- Evaluation of integrals-II
- Analytic Continuation
- Introduction of orthogonal polynomials
- How to construct orthogonal polynomials
- The weight function
- Recursion relations
- Differential equation satisfied by the orthogonal polynomials
- Hermite polynomials
- Properties of Hemite polynomials
- Legendre polynomials
- Legendre polynomials: recurrence relation
- Differential equation corresponding to Legendre polynomials
- The generating function corresponding to Legendre polynomials
- Laguerre Polynomials
- Laguerre Polynomials: recurrence relation
- Laguerre polynomials: differential equation
- Laguerre polynomials: generating function
- Bessel functions: series defination
- Bessel functions: recurrence relations
- Bessel functions: differential equation
- Bessel functions of integral order: generating function
- Bessel function: orthogonality
- Classification of Second Order PDEs
- Canonical Forms for Hyperbolic PDEs
- Canonical Forms for Parabolic PDEs
- Canonical Forms for Elliptic PDEs
- Tha Laplace Equation
- The Laplace Equation: Separation of Variables
- The Laplace Equation: Dirichlet and Neumann boundary conditions
- The Laplace Equation in Cartesian coordinates
- The Laplace Equation for a 3-D rectangular box
- The Laplace Equation in spherical coordinates
- The Laplace Equation in Spherical Coordinates: Solution
- The Laplace Equation in Spherical Coordinates: illustrative examples
- The Poisson's Equation: Green's function solution
- The heat equation: a heuristic discussion
- From the random walk to the diffusion equation
- Solution of the Diffusion equation
- The Diffusion equation with Dirichlet and Neumann boundary conditions
- The Heat equation: illustrative examples
- The Wave equation: Method of characteristics
- The Wave equation: Separation of variables