Riemann's differential equation
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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.