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 differential equation is given by
The regular singular points are a, b, and c. The pairs of exponents[clarification needed] for each are respectively α; α′, β; β′, and γ; γ′. The exponents are subject to the condition
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
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". Retrieved 21 April 2014.