§15.11 Riemann’s Differential Equation

§15.11(i) Equations with Three Singularities

The importance of (15.10.1) is that any homogeneous linear differential equation of the second order with at most three distinct singularities, all regular, in the extended plane can be transformed into (15.10.1). The most general form is given by

 15.11.1 $\frac{{d}^{2}w}{{dz}^{2}}+\left(\frac{1-a_{1}-a_{2}}{z-\alpha}+\frac{1-b_{1}-b% _{2}}{z-\beta}+\frac{1-c_{1}-c_{2}}{z-\gamma}\right)\frac{dw}{dz}+{\left(\frac% {(\alpha-\beta)(\alpha-\gamma)a_{1}a_{2}}{z-\alpha}+\frac{(\beta-\alpha)(\beta% -\gamma)b_{1}b_{2}}{z-\beta}+\frac{(\gamma-\alpha)(\gamma-\beta)c_{1}c_{2}}{z-% \gamma}\right)\frac{w}{(z-\alpha)(z-\beta)(z-\gamma)}=0},$

with

 15.11.2 $a_{1}+a_{2}+b_{1}+b_{2}+c_{1}+c_{2}=1.$

Here $\{a_{1},a_{2}\}$, $\{b_{1},b_{2}\}$, $\{c_{1},c_{2}\}$ are the exponent pairs at the points $\alpha$, $\beta$, $\gamma$, respectively. Cases in which there are fewer than three singularities are included automatically by allowing the choice $\{0,1\}$ for exponent pairs. Also, if any of $\alpha$, $\beta$, $\gamma$, is at infinity, then we take the corresponding limit in (15.11.1).

The complete set of solutions of (15.11.1) is denoted by Riemann’s $\mathop{P\/}\nolimits$-symbol:

 15.11.3 $w=\mathop{P\/}\nolimits\!\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}.$ Defines: $\mathop{P\/}\nolimits\!\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}$: Riemann’s $\mathop{P\/}\nolimits$-symbol for solutions of the generalized hypergeometric differential equation Symbols: $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter, $\alpha$: point, $\beta$: point, $\gamma$: point and $w$: solution Permalink: http://dlmf.nist.gov/15.11.E3 Encodings: TeX, pMML, png

In particular,

 15.11.4 $w=\mathop{P\/}\nolimits\!\begin{Bmatrix}0&1&\infty&\\ 0&0&a&z\\ 1-c&c-a-b&b&\end{Bmatrix}$

denotes the set of solutions of (15.10.1).

§15.11(ii) Transformation Formulas

A conformal mapping of the extended complex plane onto itself has the form

 15.11.5 $t=\ifrac{(\kappa z+\lambda)}{(\mu z+\nu)},$

where $\kappa$, $\lambda$, $\mu$, $\nu$ are real or complex constants such that $\kappa\nu-\lambda\mu=1$. These constants can be chosen to map any two sets of three distinct points $\{\alpha,\beta,\gamma\}$ and $\{\widetilde{\alpha},\widetilde{\beta},\widetilde{\gamma}\}$ onto each other. Symbolically:

 15.11.6 $\mathop{P\/}\nolimits\!\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}=\mathop{P\/}\nolimits\!\begin{Bmatrix}% \widetilde{\alpha}&\widetilde{\beta}&\widetilde{\gamma}&\\ a_{1}&b_{1}&c_{1}&t\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}.$

The reduction of a general homogeneous linear differential equation of the second order with at most three regular singularities to the hypergeometric differential equation is given by

 15.11.7 $\mathop{P\/}\nolimits\!\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}=\left(\frac{z-\alpha}{z-\gamma}\right)^{a_{1}}% \left(\frac{z-\beta}{z-\gamma}\right)^{b_{1}}\mathop{P\/}\nolimits\!\begin{% Bmatrix}0&1&\infty&\\ 0&0&a_{1}+b_{1}+c_{1}&\dfrac{(z-\alpha)(\beta-\gamma)}{(z-\gamma)(\beta-\alpha% )}\\ a_{2}-a_{1}&b_{2}-b_{1}&a_{1}+b_{1}+c_{2}&\end{Bmatrix}.$

We also have

 15.11.8 $z^{\lambda}(1-z)^{\mu}\mathop{P\/}\nolimits\!\begin{Bmatrix}0&1&\infty&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}=\mathop{P\/}\nolimits\!\begin{Bmatrix}0&1&% \infty&\\ a_{1}+\lambda&b_{1}+\mu&c_{1}-\lambda-\mu&z\\ a_{2}+\lambda&b_{2}+\mu&c_{2}-\lambda-\mu&\end{Bmatrix},$

for arbitrary $\lambda$ and $\mu$.