In mathematics, Riemann's differential equation, named after Bernhard Riemann, is a generalization of the hypergeometric differential equation, allowing the regular singular points (RSPs) 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 RSPs, while other transformations (see below) can change the exponents at the RSPs, subject to the exponents adding up to 1.
The differential equation is given by
The regular singular points are a, b, and c. The exponents of the solutions at these RSPs are, respectively, α; α′, β; β′, and γ; γ′. As before, the exponents are subject to the condition
Solutions and relationship with the hypergeometric function
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 transformations
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 AD − BC ≠ 0, define the quantities
then one has the simple relation
expressing the symmetry.
- ^ Siklos, Stephen. "The Papperitz equation" (PDF). Retrieved 21 April 2014.
- 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