Modular forms | Ordinary differential equations | Hypergeometric functions
In the following we solve the second-order differential equation called the hypergeometric differential equation using Frobenius method, named after Ferdinand Georg Frobenius. This is a method that uses the series solution for a differential equation, where we assume the solution takes the form of a series. This is usually the method we use for complicated ordinary differential equations. The solution of the hypergeometric differential equation is very important. For instance, Legendre's differential equation can be shown to be a special case of the hypergeometric differential equation. Hence, by solving the hypergeometric differential equation, one may directly compare its solutions to get the solutions of Legendre's differential equation, after making the necessary substitutions. For more details, please check the hypergeometric differential equation. We shall prove that this equation has three singularities, namely at x = 0, x = 1 and around x = infinity. However, as these will turn out to be regular singular points, we will be able to assume a solution on the form of a series. Since this is a second-order differential equation, we must have two linearly independent solutions. The problem however will be that our assumed solutions may or not be independent, or worse, may not even be defined (depending on the value of the parameters of the equation). This is why we shall study the different cases for the parameters and modify our assumed solution accordingly. (Wikipedia).
C73 Introducing the theorem of Frobenius
The theorem of Frobenius allows us to calculate a solution around a regular singular point.
From playlist Differential Equations
Differential Equations | Frobenius' Method -- Example 1
From the desert, we present an example of a Frobenius series solution to a second order homogeneous differential equation. http://www.michael-penn.net http://www.randolphcollege.edu/mathematics/
From playlist Series Solutions for Differential Equations
Differential Equations | Frobenius' Method part 2
From Garden of the Gods in Colorado Springs, we present a Theorem regarding Frobenius Series solutions to a certain family of second order homogeneous differential equations. An example is also explored. http://www.michael-penn.net http://www.randolphcollege.edu/mathematics/
From playlist Series Solutions for Differential Equations
Differential Equations | Frobenius' Method part 1
From the bridge of the Starship Enterprise, we present a Theorem which will form the basis for a Frobenius solution to a certain family of 2nd order differential equations. http://www.michael-penn.net http://www.randolphcollege.edu/mathematics/
From playlist Series Solutions for Differential Equations
Differential Equations | Frobenius' Method: Example 2
We give an example of solving a second order differential equations using Frobenius' method. http://www.michael-penn.net http://www.randolphcollege.edu/mathematics/
From playlist Series Solutions for Differential Equations
Duco van Straten: CY-motives and differential equations
conference Recorded during the meeting "D-Modules: Applications to Algebraic Geometry, Arithmetic and Mirror Symmetry" the April 12, 2022 by the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Guillaume Hennenfent Find this video and other talks given by
From playlist Algebraic and Complex Geometry
Kiran S. Kedlaya: Frobenius structures on hypergeometric equations: computational methods
The lecture was held within the framework of the Hausdorff Trimester Program: Periods in Number Theory, Algebraic Geometry and Physics. Abstract: Current implementations of the computation of L-functions associated to hypergeometric motives in Magma and Sage rely on a p-adic trace formula
From playlist HIM Lectures: Trimester Program "Periods in Number Theory, Algebraic Geometry and Physics"
Bartosz Naskręcki: Elliptic and hyperelliptic realisations of low degree hypergeometric motives
The lecture was held within the framework of the Hausdorff Trimester Program: Periods in Number Theory, Algebraic Geometry and Physics. Abstract: In this talk we will discuss what are the so-called hypergeometric motives and how one can approach the problem of their explicit construction
From playlist HIM Lectures: Trimester Program "Periods in Number Theory, Algebraic Geometry and Physics"
The Frobenius Problem - Method for Finding the Frobenius Number of Two Numbers
Goes over how to find the Frobenius Number of two Numbers.
From playlist ℕumber Theory
This is one of my all-time favorite differential equation videos!!! :D Here I'm actually using the Wronskian to actually find a nontrivial solution to a second-order differential equation. This is amazing because it brings the concept of the Wronskian back to life! And as they say, you won
From playlist Differential equations
Strong Frobenius structure, rigidity and hypergeometric equations Firstly, we will show that if L is a Fucshian differential operator with coefficients in \mathbb{Q}(z), whose monodromy group is rigid and the exponents are rational numbers at singular points, then L has a strong Frobenius
From playlist DART X
What's New in Calculus and Algebra
To learn more about Wolfram Technology Conference, please visit: https://www.wolfram.com/events/technology-conference/ Speaker: Devendra Kapadia & Adam Strzebonski Wolfram developers and colleagues discussed the latest in innovative technologies for cloud computing, interactive deploymen
From playlist Wolfram Technology Conference 2018
Hypergeometric Motives - Fernando Villegas
Fernando Villegas University Texas at Austin March 15, 2012 The families of motives of the title arise from classical one-variable hypergeometric functions. This talk will focus on the calculation of their corresponding L-functions both in theory and in practice. These L-functions provide
From playlist Mathematics
Nicholas Katz - Exponential sums and finite groups
Correction: The affiliation of Lei Fu is Tsinghua University. This is joint work with Antonio Rojas Leon and Pham Huu Tiep, where we look for “interesting” finite groups arising as monodromy groups of “simple to remember” families of exponential sums”.
From playlist Conférence « Géométrie arithmétique en l’honneur de Luc Illusie » - 5 mai 2021
Free ebook http://tinyurl.com/EngMathYT I show how to solve differential equations by applying the method of variation of parameters for those wanting to review their understanding.
From playlist Differential equations
Andrew Sutherland, Arithmetic L-functions and their Sato-Tate distributions
VaNTAGe seminar on April 28, 2020. License: CC-BY-NC-SA Closed captions provided by Jun Bo Lau.
From playlist The Sato-Tate conjecture for abelian varieties
Differential Equations | Application of Abel's Theorem Example 1
We give an example of applying Abel's Theorem to construct a second solution to a differential equation given one solution. www.michael-penn.net
From playlist Differential Equations
Power series solution to differential equations: a tutorial
Free ebook http://tinyurl.com/EngMathYT How to solve differential equations using power series. An example is discussed involving the method of Frobenius where linear differential equation (with variable coefficients) is solved by using a power series. The ideas are seen in university
From playlist A second course in university calculus.
Thin Matrix Groups - a brief survey of some aspects - Peter Sarnak
Speaker: Peter Sarnak (Princeton/IAS) Title: Thin Matrix Groups - a brief survey of some aspects More videos on http://video.ias.edu
From playlist Mathematics