Ordinary differential equations | Hypergeometric functions

Riemann's differential equation

In mathematics, Riemann's differential equation, named after Bernhard Riemann, is a generalization of the hypergeometric differential equation, allowing the regular singular points to occur anywhere on the Riemann sphere, rather than merely at 0, 1, and . The equation is also known as the Papperitz equation. The hypergeometric differential equation is a second-order linear differential equation which has three regular singular points, 0, 1 and . That equation admits two linearly independent solutions; near a singularity , the solutions take the form , where is a local variable, and is locally holomorphic with . The real number is called the exponent of the solution at . Let α, β and γ be the exponents of one solution at 0, 1 and respectively; and let α', β' and γ' be those of the other. Then By applying suitable changes of variable, it is possible to transform the hypergeometric equation: Applying Möbius transformations will adjust the positions of the regular singular points, while other transformations (see below) can change the exponents at the regular singular points, subject to the exponents adding up to 1. (Wikipedia).

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The theorem of Frobenius allows us to calculate a solution around a regular singular point.

From playlist Differential Equations

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I show how the Definition of Area of a Plane is a special case of the Riemann Sum. When finding the area of a plane bound by a function and an axis on a closed interval, the width of the partitions (probably rectangles) does not have to be equal. I work through two examples that are rela

From playlist Calculus

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Solve a Bernoulli Differential Equation (Part 2)

This video provides an example of how to solve an Bernoulli Differential Equation. The solution is verified graphically. Library: http://mathispower4u.com

From playlist Bernoulli Differential Equations

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Learn how to solve the particular solution of differential equations. A differential equation is an equation that relates a function with its derivatives. The solution to a differential equation involves two parts: the general solution and the particular solution. The general solution give

From playlist Differential Equations

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I introduce the Definite Integral, explain the definition, and work through an example of using the Definite Integral notation and Riemann Sum to find the area bound by a function and the x axis on a closed interval. Find free review test, useful notes and more at http://www.mathplane.com

From playlist Calculus

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Learn how to solve the particular solution of differential equations. A differential equation is an equation that relates a function with its derivatives. The solution to a differential equation involves two parts: the general solution and the particular solution. The general solution give

From playlist Solve Differential Equation (Particular Solution) #Integration

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From playlist Riemann Sum Approximation

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See how to solve a Bernoulli equation.

From playlist Differential Equations

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In this video I explain Riemann's zeta function and the Riemann hypothesis. I also implement and algorithm to compute the return values - here's the Python script:https://gist.github.com/Nikolaj-K/996dba1ff1045d767b10d4d07b1b032f

From playlist Programming

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Complex differentiable functions, the Cauchy-Riemann equations and an application.

From playlist MATH2069 Complex Analysis

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At the start of the 20th century, David Hilbert asked which representations can arise by studying the monodromy of Fuchsian equations. This question was the starting point for a beautiful circle of ideas relating the topology of a complex algebraic variety X to the study of algebraic diffe

From playlist Felix Klein Lectures 2022

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From playlist Integrable​ ​systems​ ​in​ ​Mathematics,​ ​Condensed​ ​Matter​ ​and​ ​Statistical​ ​Physics

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From playlist Engineering Math: Crash Course in Complex Analysis

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From playlist An Introduction to the Arithmetic of Elliptic Curves

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Please Subscribe here, thank you!!! https://goo.gl/JQ8Nys Using the Cauchy-Riemann equations to prove that the function f(z) = Rez is nowhere differentiable. This is a straight forward application of the C.R. equations.

From playlist Complex Analysis

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Complex Analysis - Part 6 - Cauchy-Riemann Equations

Support the channel on Steady: https://steadyhq.com/en/brightsideofmaths Or support me via PayPal: https://paypal.me/brightmaths Or via Ko-fi: https://ko-fi.com/thebrightsideofmathematics Or via Patreon: https://www.patreon.com/bsom Or via other methods: https://thebrightsideofmathematics.

From playlist Complex Analysis

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From playlist Integrable​ ​systems​ ​in​ ​Mathematics,​ ​Condensed​ ​Matter​ ​and​ ​Statistical​ ​Physics

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Limits of cubic differentials and projective structures by David Dumas

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From playlist Surface Group Representations and Geometric Structures

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Claudia Fevola - KP Solitons from Tropical Limits

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From playlist Research Spotlight

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From playlist Solve Differential Equation (Particular Solution) #Integration

Related pages

Monodromy | Möbius transformation | Abramowitz and Stegun | Riemann sphere | Hypergeometric function | Bernhard Riemann | Complex number | Ernst Kummer | Mathematics | General linear group | Conformal map