Riemann's differential equation
Encyclopedia
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 ∞.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Riemann's differential equation, named after Bernhard Riemann
Bernhard Riemann
Georg Friedrich Bernhard Riemann was an influential German mathematician who made lasting contributions to analysis and differential geometry, some of them enabling the later development of general relativity....
, is a generalization of the hypergeometric differential equation
Hypergeometric differential equation
In mathematics, the Gaussian or ordinary hypergeometric function 2F1 is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. It is a solution of a second-order linear ordinary differential equation...
, allowing the regular singular points to occur anywhere on the Riemann sphere
Riemann sphere
In mathematics, the Riemann sphere , named after the 19th century mathematician Bernhard Riemann, is the sphere obtained from the complex plane by adding a point at infinity...
, rather than merely at 0,1, and ∞.
Definition
The differential equation is given by
-
The regular singular points are a, b and c. The pairs of exponents for each are respectively α; α', β;β' and γ;γ'. The exponents are subject to the condition
Solutions
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 equationHypergeometric differential equationIn mathematics, the Gaussian or ordinary hypergeometric function 2F1 is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. It is a solution of a second-order linear ordinary 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, that is, under the action of the group GL(2, C), or equivalently, under the conformal remappingConformal mapIn mathematics, a conformal map is a function which preserves angles. In the most common case the function is between domains in the complex plane.More formally, a map,...
s of the Riemann sphereRiemann sphereIn mathematics, the Riemann sphere , named after the 19th century mathematician Bernhard Riemann, is the sphere obtained from the complex plane by adding a point at infinity...
. Given arbitrary complex numberComplex numberA complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s A, B, C, D such that AD − BC ≠ 0, define the quantities
and
then one has the simple relation
expressing the symmetry.