Riemann's differential equation
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In mathematics, Riemann's differential equation 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 ∞.
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[edit] Definition
The differential equation is given by
![\frac{d^2w}{dz^2} + \left[
\frac{1-\alpha-\alpha'}{z-a} +
\frac{1-\beta-\beta'}{z-b} +
\frac{1-\gamma-\gamma'}{z-c} \right] \frac{dw}{dz}](../../../../math/9/8/9/98909a4101fc979c3ca83fb5f72b5b9f.png)
![+\left[
\frac{\alpha\alpha' (a-b)(a-c)} {z-a}
+\frac{\beta\beta' (b-c)(b-a)} {z-b}
+\frac{\gamma\gamma' (c-a)(c-b)} {z-c}
\right]
\frac{w}{(z-a)(z-b)(z-c)}=0.](../../../../math/b/4/9/b4945a13dcd293dc64b94cfb906dd3e5.png)
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
- α + α' + β + β' + γ + γ' = 1.
[edit] 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 equation for a treatment of Kummer's solutions.
[edit] 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 remappings of the Riemann sphere. Given arbitrary complex numbers A,B,C,D such that 
, define the quantities
and
then one has the simple relation
expressing the symmetry.
[edit] References
- 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
- Chapter 15 Hypergeometric Functions
