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
infty
The hypergeometric differential equation is a second-order linear differential equation which has three regular singular points, 0, 1 and
infty
zs
xsf(x)
x=z-zs
f
f(0) ≠ 0
s
zs
infty
\alpha+\alpha'+\beta+\beta'+\gamma+\gamma'=1.
The differential equation is given by
| d2w | |
| dz2 |
+\left[
| 1-\alpha-\alpha' | + | |
| z-a |
| 1-\beta-\beta' | + | |
| z-b |
| 1-\gamma-\gamma' | |
| z-c |
\right]
| dw | |
| dz |
| +\left[ | \alpha\alpha'(a-b)(a-c) | + |
| z-a |
| \beta\beta'(b-c)(b-a) | + | |
| z-b |
| \gamma\gamma'(c-a)(c-b) | \right] | |
| z-c |
| w | |
| (z-a)(z-b)(z-c) |
=0.
The regular singular points are,, and . The exponents of the solutions at these regular singular points are, respectively,,, and . As before, the exponents are subject to the condition
\alpha+\alpha'+\beta+\beta'+\gamma+\gamma'=1.
The solutions are denoted by the Riemann P-symbol (also known as the Papperitz symbol)
w(z)=P\left\{\begin{matrix}a&b&c& \ \alpha&\beta&\gamma&z\\ \alpha'&\beta'&\gamma'& \end{matrix}\right\}
The standard hypergeometric function may be expressed as
2F1(a,b;c;z)= P\left\{\begin{matrix}0&infty&1& \ 0&a&0&z\\ 1-c&b&c-a-b& \end{matrix}\right\}
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
P\left\{\begin{matrix}a&b&c& \ \alpha&\beta&\gamma&z\\ \alpha'&\beta'&\gamma'& \end{matrix}\right\}=\left(
| z-a | |
| z-b |
\right)\alpha\left(
| z-c | |
| z-b |
\right)\gamma P\left\{\begin{matrix}0&infty&1& \ 0&\alpha+\beta+\gamma&0&
| (z-a)(c-b) | |
| (z-b)(c-a) |
\\ \alpha'-\alpha&\alpha+\beta'+\gamma&\gamma'-\gamma& \end{matrix}\right\}
In other words, one may write the solutions in terms of the hypergeometric function as
| w(z)= \left( | z-a |
| z-b |
\right)\alpha\left(
| z-c | |
| z-b |
| \gamma | |
| \right) | |
| 2F |
1\left(\alpha+\beta+\gamma,\alpha+\beta'+\gamma;1+\alpha-\alpha';
| (z-a)(c-b) | |
| (z-b)(c-a) |
\right)
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.
The P-function possesses a simple symmetry under the action of fractional linear transformations known as Möbius transformations (that are the conformal remappings of the Riemann sphere), or equivalently, under the action of the group . Given arbitrary complex numbers,,, such that, define the quantities
| u= | Az+B |
| Cz+D |
and η=
| Aa+B | |
| Ca+D |
and
| \zeta= | Ab+B |
| Cb+D |
and \theta=
| Ac+B | |
| Cc+D |
then one has the simple relation
P\left\{\begin{matrix}a&b&c& \ \alpha&\beta&\gamma&z\\ \alpha'&\beta'&\gamma'& \end{matrix}\right\} =P\left\{\begin{matrix}η&\zeta&\theta& \ \alpha&\beta&\gamma&u\\ \alpha'&\beta'&\gamma'& \end{matrix}\right\}
expressing the symmetry.
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:[2] [3]
| \left( | z-a |
| z-b |
| ||||
| \right) |
\right)lP\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\}