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{{Short description|Generalization of the hypergeometric 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 <math>\infty</math>. The equation is also known as the '''Papperitz equation'''.<ref>{{cite web|last=Siklos|first=Stephen|title=The Papperitz equation|url=http://www.damtp.cam.ac.uk/user/stcs/courses/fcm/handouts/papperitz.pdf|accessdate=21 April 2014|archive-date=4 March 2016|archive-url=https://web.archive.org/web/20160304030539/http://www.damtp.cam.ac.uk/user/stcs/courses/fcm/handouts/papperitz.pdf|url-status=dead}}</ref> 

The [[hypergeometric differential equation]] is a second-order linear differential equation which has three regular singular points, 0, 1 and <math>\infty</math>. That equation admits two linearly independent solutions; near a singularity <math>z_s</math>, the solutions take the form <math>x^s f(x)</math>, where <math>x = z-z_s</math> is a local variable, and <math>f</math> is locally holomorphic with <math>f(0)\neq0</math>. The real number <math>s</math> is called the exponent of the solution at <math>z_s</math>. Let ''α'', ''&beta;'' and ''&gamma;'' be the exponents of one solution at 0, 1 and <math>\infty</math> respectively; and let ''{{prime|α}}'', ''{{prime|β}}'' and ''{{prime|γ}}'' be those of the other. Then
:<math>\alpha + \alpha' + \beta + \beta' + \gamma + \gamma' = 1.</math>
By applying suitable changes of variable, it is possible to transform the hypergeometric equation: Applying [[Möbius transformation]]s 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.

==Definition==
The differential equation is given by
:<math>\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>
::<math>+\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>

The regular singular points are {{mvar|a}}, {{mvar|b}}, and {{mvar|c}}. The exponents of the solutions at these regular singular points are, respectively, {{math|''α''; ''{{prime|α}}''}}, {{math|''β''; ''{{prime|β}}''}}, and {{math|''γ''; ''{{prime|γ}}''}}. As before, the exponents are subject to the condition
:<math>\alpha+\alpha'+\beta+\beta'+\gamma+\gamma'=1.</math>

==Solutions and relationship with the hypergeometric function==
The solutions are denoted by the ''Riemann P-symbol'' (also known as the ''Papperitz symbol'')

:<math>w(z)=P  \left\{ \begin{matrix} a & b & c & \; \\ 
\alpha & \beta & \gamma & z \\
\alpha' & \beta' & \gamma' & \;
\end{matrix} \right\}</math>

The standard [[hypergeometric function]] may be expressed as

:<math>\;_2F_1(a,b;c;z) =
P  \left\{ \begin{matrix} 0 & \infty & 1 & \; \\ 
0 & a & 0 & z \\
1-c & b & c-a-b & \;
\end{matrix} \right\}</math>

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

:<math>P  \left\{ \begin{matrix} a & b & c & \; \\ 
\alpha & \beta & \gamma & z \\
\alpha' & \beta' & \gamma' & \;
\end{matrix} \right\} = 
\left(\frac{z-a}{z-b}\right)^\alpha 
\left(\frac{z-c}{z-b}\right)^\gamma
P \left\{ \begin{matrix} 0 & \infty & 1 & \; \\ 
0 & \alpha+\beta+\gamma & 0 & \;\frac{(z-a)(c-b)}{(z-b)(c-a)} \\
\alpha'-\alpha & \alpha+\beta'+\gamma & \gamma'-\gamma & \;
\end{matrix} \right\}
</math>

In other words, one may write the solutions in terms of the hypergeometric function as

:<math>w(z)=
\left(\frac{z-a}{z-b}\right)^\alpha 
\left(\frac{z-c}{z-b}\right)^\gamma
\;_2F_1 \left(
\alpha+\beta +\gamma, 
\alpha+\beta'+\gamma; 
1+\alpha-\alpha';
\frac{(z-a)(c-b)}{(z-b)(c-a)} \right) 
</math>

The full complement of [[Ernst Kummer|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 known as [[Möbius transformation]]s (that are the [[conformal map|conformal remappings]] of the Riemann sphere), or equivalently, under the action of the group {{math|[[general linear group|GL]](2, '''C''')}}. Given arbitrary [[complex number]]s {{mvar|A}}, {{mvar|B}}, {{mvar|C}}, {{mvar|D}} such that {{math|''AD'' − ''BC'' ≠ 0}}, define the quantities

:<math>u=\frac{Az+B}{Cz+D} 
\quad \text{ and } \quad
\eta=\frac{Aa+B}{Ca+D}</math>

and

:<math>\zeta=\frac{Ab+B}{Cb+D}
\quad \text{ and } \quad
\theta=\frac{Ac+B}{Cc+D}</math>

then one has the simple relation

:<math>P  \left\{ \begin{matrix} a & b & c & \; \\ 
\alpha & \beta & \gamma & z \\
\alpha' & \beta' & \gamma' & \;
\end{matrix} \right\}
=P  \left\{ \begin{matrix} 
\eta & \zeta & \theta & \; \\ 
\alpha & \beta & \gamma & u \\
\alpha' & \beta' & \gamma' & \;
\end{matrix} \right\}</math>

expressing the symmetry.
== Exponents ==
If the Moebius transformation above moves the singular points but does not change the exponents, 
the following transformation does not move the singular points but changes the exponents:
<ref>{{cite book|last=Whittaker|title=A course in modern analysis|url=https://archive.org/details/courseofmodernan00whit|accessdate=30 September 2021|section=10.7,14.2|pages= 201,277}}</ref>
<ref>{{cite web | title=The Hypergeometric Function and the Papperitz Equation
|author=Richard Chapling| url=https://rc476.user.srcf.net/furthercomplexmethods/hypergeometric.pdf|accessdate=30 September 2021}}</ref>
:<math>\left(\frac{z-a}{z-b}\right)^k\left(\frac{z-c}{z-b}\right)^l P  \left\{ \begin{matrix} a & b & c & \; \\ 
\alpha & \beta  & \gamma & z \\
\alpha' & \beta' & \gamma' & \;
\end{matrix} \right\}
=P  \left\{ \begin{matrix} 
a & b & c & \; \\ 
\alpha +k & \beta -k -l & \gamma + l & z \\
\alpha' +k & \beta'-k -l & \gamma' + l & \;
\end{matrix} \right\}</math>

==See also==
*[[Method of Frobenius]]
*[[Monodromy]]

==Notes==
{{reflist}}

==References==
* Milton Abramowitz and Irene A. Stegun, eds., ''[[Abramowitz and Stegun|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

{{series (mathematics)}}
{{Bernhard Riemann}}

[[Category:Hypergeometric functions]]
[[Category:Ordinary differential equations]]
[[Category:Bernhard Riemann]]