Riemann's differential equation

The differential equation is given by

${\displaystyle {\frac {d^{2}w}{dz^{2}}}+\left[{\frac {1-\alpha -\alpha '}{z-a}}+{\frac {1-\beta -\beta '}{z-b}}+{\frac {1-\gamma -\gamma '}{z-c}}\right]{\frac {dw}{dz}}}$

${\displaystyle +\left[{\frac {\alpha \alpha '(a-b)(a-c)}{z-a}}+{\frac {\beta \beta '(b-c)(b-a)}{z-b}}+{\frac {\gamma \gamma '(c-a)(c-b)}{z-c}}\right]{\frac {w}{(z-a)(z-b)(z-c)}}=0.}$

The regular singular points are a, b, and c. The exponents of the solutions at these RSPs are, respectively, α; α′, β; β′, and γ; γ′. As before, the exponents are subject to the condition

${\displaystyle \alpha +\alpha '+\beta +\beta '+\gamma +\gamma '=1.}$

The solutions are denoted by the Riemann P-symbol (also known as the Papperitz symbol)

${\displaystyle w(z)=P\left\{{\begin{matrix}a&b&c&\;\\\alpha &\beta &\gamma &z\\\alpha '&\beta '&\gamma '&\;\end{matrix}}\right\}}$

The standard hypergeometric function may be expressed as

${\displaystyle \;_{2}F_{1}(a,b;c;z)=P\left\{{\begin{matrix}0&\infty &1&\;\\0&a&0&z\\1-c&b&c-a-b&\;\end{matrix}}\right\}}$

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

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

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

${\displaystyle w(z)=\left({\frac {z-a}{z-b}}\right)^{\alpha }\left({\frac {z-c}{z-b}}\right)^{\gamma }\;_{2}F_{1}\left(\alpha +\beta +\gamma ,\alpha +\beta '+\gamma ;1+\alpha -\alpha ';{\frac {(z-a)(c-b)}{(z-b)(c-a)}}\right)}$

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 GL(2, C). Given arbitrary complex numbers A, B, C, D such that AD − BC ≠ 0, define the quantities

${\displaystyle u={\frac {Az+B}{Cz+D}}\quad {\text{ and }}\quad \eta ={\frac {Aa+B}{Ca+D}}}$

and

${\displaystyle \zeta ={\frac {Ab+B}{Cb+D}}\quad {\text{ and }}\quad \theta ={\frac {Ac+B}{Cc+D}}}$

then one has the simple relation

${\displaystyle P\left\{{\begin{matrix}a&b&c&\;\\\alpha &\beta &\gamma &z\\\alpha '&\beta '&\gamma '&\;\end{matrix}}\right\}=P\left\{{\begin{matrix}\eta &\zeta &\theta &\;\\\alpha &\beta &\gamma &u\\\alpha '&\beta '&\gamma '&\;\end{matrix}}\right\}}$

expressing the symmetry.

• Complex differential equation
• Method of Frobenius
• Monodromy

• 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