§15.11 Riemann’s Differential Equation

Keywords:: Riemann’s differential equation
Referenced by:: §15.17(i)
Permalink:: http://dlmf.nist.gov/15.11

§15.11(i) Equations with Three Singularities
§15.11(ii) Transformation Formulas

§15.11(i) Equations with Three Singularities

Notes:: See Andrews et al. (1999, §2.3) or Olver (1997b, pp. 156–158).
A&S Ref:: 15.6
Keywords:: Riemann’s differential equation, Riemann’s $\mathop{P\/}\nolimits$ -symbol, singularities
Referenced by:: ¶ ‣ §31.10(i), ¶ ‣ §31.10(ii), §31.11(ii)
Permalink:: http://dlmf.nist.gov/15.11.i

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

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $z$ : 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
Referenced by:: §15.11(i), §15.11(i), §15.17(v)
Permalink:: http://dlmf.nist.gov/15.11.E1
Encodings:: TeX, pMML, png

with

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

Symbols:: $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/15.11.E2
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{P\/}\nolimits\!\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}$ : Riemann’s $\mathop{P\/}\nolimits$ -symbol for solutions of the generalized hypergeometric differential equation, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter and $w$ : solution
Permalink:: http://dlmf.nist.gov/15.11.E4
Encodings:: TeX, pMML, png

denotes the set of solutions of (15.10.1).

§15.11(ii) Transformation Formulas

Notes:: See Andrews et al. (1999, §2.3) or Olver (1997b, pp. 156–158).
A&S Ref:: 15.6
Keywords:: Riemann’s differential equation
Permalink:: http://dlmf.nist.gov/15.11.ii

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

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

Symbols:: $z$ : complex variable, $t$ : mapping, $\kappa$ : constant, $\lambda$ : constant, $\mu$ : constant and $\nu$ : constant
Permalink:: http://dlmf.nist.gov/15.11.E5
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{P\/}\nolimits\!\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}$ : Riemann’s $\mathop{P\/}\nolimits$ -symbol for solutions of the generalized hypergeometric differential equation, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter, $\alpha$ : point, $\beta$ : point, $\gamma$ : point and $t$ : mapping
Permalink:: http://dlmf.nist.gov/15.11.E6
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{P\/}\nolimits\!\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}$ : Riemann’s $\mathop{P\/}\nolimits$ -symbol for solutions of the generalized hypergeometric differential equation, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter, $\alpha$ : point, $\beta$ : point and $\gamma$ : point
Permalink:: http://dlmf.nist.gov/15.11.E7
Encodings:: TeX, pMML, png

We also have

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

Symbols:: $\mathop{P\/}\nolimits\!\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}$ : Riemann’s $\mathop{P\/}\nolimits$ -symbol for solutions of the generalized hypergeometric differential equation, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter, $\lambda$ : constant and $\mu$ : constant
Permalink:: http://dlmf.nist.gov/15.11.E8
Encodings:: TeX, pMML, png

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

§15.11 Riemann’s Differential Equation

Contents

§15.11(i) Equations with Three Singularities

§15.11(ii) Transformation Formulas