- Riemann's differential equation
In
mathematics , Riemann's differential equation is a generalization of thehypergeometric differential equation , allowing theregular singular points to occur anywhere on theRiemann sphere , 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
exponent s for each are respectively α; α', β;β' and γ;γ'. The exponents are subject to the condition:
olutions
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 transformation s, that is, under the action of the group GL(2, C), or equivalently, under the conformal remappings of theRiemann sphere . Given arbitrarycomplex number s "A, B, C, D" such that "AD" − "BC" ne; 0, define the quantities:
and
:
then one has the simple relation
:
expressing the symmetry.
ee also
*
Complex differential equation References
* Milton Abramowitz and Irene A. Stegun, eds., "
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables" (Dover: New York, 1972)
** [http://www.math.sfu.ca/~cbm/aands/page_556.htm Chapter 15] Hypergeometric Functions
*** [http://www.math.sfu.ca/~cbm/aands/page_564.htm Section 15.6] Riemann's Differential Equation
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