Riemann's differential equation

Summary

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.[1]

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.

DefinitionEdit

The differential equation is given by

 
 

The regular singular points are a, b, and c. The exponents of the solutions at these regular singular points are, respectively, α; α′, β; β′, and γ; γ′. As before, the exponents are subject to the condition

 

Solutions and relationship with the hypergeometric functionEdit

The solutions are denoted by the Riemann P-symbol (also known as the Papperitz symbol)

 

The standard hypergeometric function may be expressed as

 

The P-functions obey a number of identities; one of them allows a general P-function to be expressed in terms of the hypergeometric function. It is

 

In other words, one may write the solutions in terms of the hypergeometric function as

 

The full complement of Kummer's 24 solutions may be obtained in this way; see the article hypergeometric differential equation for a treatment of Kummer's solutions.

Fractional linear transformationsEdit

The P-function possesses a simple symmetry under the action of fractional linear transformations known as Möbius transformations (that are the conformal remappings of the Riemann sphere), or equivalently, under the action of the group GL(2, C). Given arbitrary complex numbers A, B, C, D such that ADBC ≠ 0, define the quantities

 

and

 

then one has the simple relation

 

expressing the symmetry.

ExponentsEdit

If the Moebius transformation above moves the singular points but does not change the exponents, the following transformation does not move the singular points but changes the exponents: [2][3]

 

See alsoEdit

NotesEdit

  1. ^ Siklos, Stephen. "The Papperitz equation" (PDF). Retrieved 21 April 2014.
  2. ^ Whittaker. "10.7,14.2". A course in modern analysis. pp. 201, 277. Retrieved 30 September 2021.
  3. ^ Richard Chapling. "The Hypergeometric Function and the Papperitz Equation" (PDF). Retrieved 30 September 2021.

ReferencesEdit

  • Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover: New York, 1972)
    • Chapter 15 Hypergeometric Functions
      • Section 15.6 Riemann's Differential Equation