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 RSPs, while other transformations can change the exponents at the RSPs, subject to the exponents adding up to 1.

Definition

The differential equation is given by
The regular singular points are,, and. The exponents of the solutions at these RSPs are, respectively,,, and. 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
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, or equivalently, under the action of the group. Given arbitrary complex numbers,,, such that, define the quantities
and
then one has the simple relation
expressing the symmetry.