Difference between revisions of "Papperitz equation"

Revision as of 14:52, 7 June 2020

An ordinary second-order Fuchsian linear differential equation having precisely three singular points:

(1)

here are pairwise distinct complex numbers, ( and ) are the characteristic exponents at the singular point (respectively, and ). A Papperitz equation is uniquely determined by the assignment of the singular points and the characteristic exponents. In solving a Papperitz equation (1), use is made of Riemann's notation:

B. Riemann investigated [1] the problem of finding all many-valued functions , analytic in the extended complex plane, which have the following properties:

a) the function has precisely three singular points ;

b) any three of its branches are connected by a linear equation

with constant coefficients;

c) the function has the simplest singularities at the points ; namely, in a neighbourhood of the point there are two branches and satisfying

where is holomorphic at ; and analogously for and .

Riemann, under certain additional assumptions on the numbers , showed that all such functions can be expressed in terms of hypergeometric functions and that satisfies a linear second-order differential equation with rational coefficients, but did not write this equation out explicitly (see [1]). The equation in question, (1), was given by E. Papperitz [2]. It is also called the Riemann -equation, the Riemann equation in Papperitz's form and the Riemann equation, and its solutions are called -functions.

The basic properties of the solutions of a Papperitz equation are as follows.

1) A Papperitz equation is invariant under rational-linear transformations: If maps the points to points , then

2) The transformation

transforms a Papperitz equation into a Papperitz equation with the same singular points, but with different characteristic exponents:

3) The hypergeometric equation

is a special case of a Papperitz equation and it corresponds in Riemann's notation to

4) Each solution of a Papperitz equation can be expressed in terms of the hypergeometric function,

(2)

under the assumption that is not a negative integer. If none of the differences , , are integers, then interchanging in (2) the positions of and or of and , four solutions of a Papperitz equation are obtained. In addition a Papperitz equation remains unchanged if the positions of the triples , , are rearranged; all these rearrangements provide 24 special solutions of a Papperitz equation (1), which were first obtained by E.E. Kummer [5].

References

[1]	B. Riemann, "Beiträge zur Theorie der durch Gauss'sche Reihe darstellbare Functionen" , Gesammelte math. Werke , Dover, reprint (1953) pp. 67–85
[2]	E. Papperitz, "Ueber verwandte -Functionen" Math. Ann. , 25 (1885) pp. 212–221 MR1510304
[3]	E.T. Whittaker, G.N. Watson, "A course of modern analysis" , Cambridge Univ. Press (1952) pp. Chapt. 6 MR1424469 MR0595076 MR0178117 MR1519757 Zbl 0951.30002 Zbl 0108.26903 Zbl 0105.26901 Zbl 53.0180.04 Zbl 47.0190.17 Zbl 45.0433.02 Zbl 33.0390.01
[4]	V.V. Golubev, "Vorlesungen über Differentialgleichungen im Komplexen" , Deutsch. Verlag Wissenschaft. (1958) (Translated from Russian) MR0100119
[5]	E.E. Kummer, "Ueber die hypergeometrische Reihe " J. Reine Angew. Math. , 15 (1836) pp. 39–83; 127–172

How to Cite This Entry:
Papperitz equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Papperitz_equation&oldid=48101

This article was adapted from an original article by M.V. Fedoryuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article

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 An ordinary second-order Fuchsian linear differential equation having precisely three singular points:
-$$ \tag{1 }
+<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p0711301.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
-w ^ {\prime\prime } + \left (
-\frac{1 - \alpha - \alpha  ^  \prime  }{z-}
- a +
-\frac{1 - \beta - \beta  ^  \prime
- }{z-}
- b +
-\frac{1- \gamma - \gamma  ^  \prime  }{z-}
- c \right ) w  ^  \prime  +
-$$
-$$
-+
-\left [
-\frac{\alpha \alpha  ^  \prime  ( a- b)( a- c) }{z-}
- a
-+
-\frac{\beta \beta  ^  \prime  ( b- c)( b- a) }{z-}
- b\right . +
-$$
-$$
+<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p0711302.png" /></td> </tr></table>
-+ \left .
-\frac{\gamma \gamma  ^  \prime  ( c- a)( c- b) }{z-}
+<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p0711303.png" /></td> </tr></table>
- c \right ]
-\frac{w}{(}
- z- a)( z- b)( z- c)  =  0 ,
-$$
-$$
+<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p0711304.png" /></td> </tr></table>
-\alpha + \alpha  ^  \prime  + \beta + \beta  ^  \prime  + \gamma + \gamma  ^  \prime   =  1;
-$$
-here  $  a, b, c $
+here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p0711305.png" /> are pairwise distinct complex numbers, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p0711306.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p0711307.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p0711308.png" />) are the characteristic exponents at the singular point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p0711309.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113011.png" />). A Papperitz equation is uniquely determined by the assignment of the singular points and the characteristic exponents. In solving a Papperitz equation (1), use is made of Riemann's notation:
-are pairwise distinct complex numbers,  $  \alpha , \alpha  ^  \prime  $(
-$  \beta , \beta  ^  \prime  $
-and  $  \gamma , \gamma  ^  \prime  $)
-are the characteristic exponents at the singular point  $  z= a $(
-respectively,  $  z= b $
-and  $  z= c $).
-A Papperitz equation is uniquely determined by the assignment of the singular points and the characteristic exponents. In solving a Papperitz equation (1), use is made of Riemann's notation:
-$$
+<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113012.png" /></td> </tr></table>
-w  =  P \left \{
-B. Riemann investigated [[#References|[1]]] the problem of finding all many-valued functions  $  w( z) $,
+B. Riemann investigated [[#References|[1]]] the problem of finding all many-valued functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113013.png" />, analytic in the extended complex plane, which have the following properties:
-analytic in the extended complex plane, which have the following properties:
-a) the function  $  w( z) $
+a) the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113014.png" /> has precisely three singular points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113015.png" />;
-has precisely three singular points  $  a, b, c $;
 b) any three of its branches are connected by a linear equation
-$$
+<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113016.png" /></td> </tr></table>
-A _ {1} w _ {1} ( z) + A _ {2} w _ {2} ( z) + A _ {3} w _ {3} ( z)  =  0
-$$
 with constant coefficients;
-c) the function  $  w( z) $
+c) the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113017.png" /> has the simplest singularities at the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113018.png" />; namely, in a neighbourhood of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113019.png" /> there are two branches <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113021.png" /> satisfying
-has the simplest singularities at the points  $  a, b, c $;
-namely, in a neighbourhood of the point  $  z= a $
-there are two branches  $  \widetilde{w}  _ {1} ( z) $
-and  $  \widetilde{w}  _ {2} ( z) $
-satisfying
-$$
+<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113022.png" /></td> </tr></table>
-\widetilde{w}  _ {1} ( z)  = \
-( z- a)  ^  \alpha  \phi _ {1} ( z) ,\ \
-\widetilde{w}  _ {2} ( z)  = \
-( z- a) ^ {\alpha  ^  \prime  } \phi _ {2} ( z) ,
-$$
-where  $  \phi _ {j} ( z) $
+where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113023.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113024.png" /> is holomorphic at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113025.png" />; and analogously for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113027.png" />.
-$  ( j = 1, 2) $
-is holomorphic at  $  z= a $;
-and analogously for  $  b $
-and  $  c $.
-Riemann, under certain additional assumptions on the numbers  $  \alpha , \alpha  ^  \prime  \dots \gamma  ^  \prime  $,
+Riemann, under certain additional assumptions on the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113028.png" />, showed that all such functions can be expressed in terms of hypergeometric functions and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113029.png" /> satisfies a linear second-order differential equation with rational coefficients, but did not write this equation out explicitly (see [[#References|[1]]]). The equation in question, (1), was given by E. Papperitz [[#References|[2]]]. It is also called the Riemann <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113031.png" />-equation, the Riemann equation in Papperitz's form and the Riemann equation, and its solutions are called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113033.png" />-functions.
-showed that all such functions can be expressed in terms of hypergeometric functions and that  $  w( z) $
-satisfies a linear second-order differential equation with rational coefficients, but did not write this equation out explicitly (see [[#References|[1]]]). The equation in question, (1), was given by E. Papperitz [[#References|[2]]]. It is also called the Riemann  $  P $-
-equation, the Riemann equation in Papperitz's form and the Riemann equation, and its solutions are called  $  P $-
-functions.
 The basic properties of the solutions of a Papperitz equation are as follows.
-) A Papperitz equation is invariant under rational-linear transformations: If  $  z _ {1} = ( Az + b)/( Cz + D) $
+) A Papperitz equation is invariant under rational-linear transformations: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113034.png" /> maps the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113035.png" /> to points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113036.png" />, then
-maps the points  $  a, b, c $
-to points  $  a _ {1} , b _ {1} , c _ {1} $,
-then
-$$
+<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113037.png" /></td> </tr></table>
-P \left \{
 ) The transformation
-$$
+<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113038.png" /></td> </tr></table>
-\left ( z-
-\frac{a}{z-}
- b \right )  ^ {k}
-\left ( z-
-\frac{c}{z-}
- b \right )  ^ {l} w  =  \widetilde{w}
-$$
 transforms a Papperitz equation into a Papperitz equation with the same singular points, but with different characteristic exponents:
-$$
+<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113039.png" /></td> </tr></table>
-\left ( z-
-\frac{a}{z-}
- b \right )  ^ {k}
-\left ( z-
-\frac{c}{z-}
- b \right )  ^ {l} P \left \{
-$$
+<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113040.png" /></td> </tr></table>
-= \
-P \left \{
 ) The [[Hypergeometric equation|hypergeometric equation]]
-$$
+<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113041.png" /></td> </tr></table>
-z( 1- z) w  ^ {\prime\prime} + [ C - ( A+ B+ 1) z] w  ^  \prime  - ABw  =  0
-$$
 is a special case of a Papperitz equation and it corresponds in Riemann's notation to
-$$
+<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113042.png" /></td> </tr></table>
-P \left \{
 ) Each solution of a Papperitz equation can be expressed in terms of the hypergeometric function,
-$$ \tag{2 }
+<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113043.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
-w( z)  =  \left ( z-
-\frac{a}{z-}
- b \right )  ^  \alpha  \left ( z-
-\frac{c}{z-}
- b \right )  ^  \gamma  \times
-$$
-$$
+<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113044.png" /></td> </tr></table>
-\times
-F \left \{ \alpha + \beta + \gamma ;  \alpha + \beta
- ^  \prime  + \gamma ;  1 + \alpha -
-\alpha  ^  \prime  ;
-\frac{( z- a)( c- b) }{( z- b)( c- a) }
- \right \}
-$$
-under the assumption that  $  \alpha - \alpha  ^  \prime  $
+under the assumption that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113045.png" /> is not a negative integer. If none of the differences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113047.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113048.png" /> are integers, then interchanging in (2) the positions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113049.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113050.png" /> or of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113051.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113052.png" />, four solutions of a Papperitz equation are obtained. In addition a Papperitz equation remains unchanged if the positions of the triples <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113053.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113054.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113055.png" /> are rearranged; all these rearrangements provide 24 special solutions of a Papperitz equation (1), which were first obtained by E.E. Kummer [[#References|[5]]].
-is not a negative integer. If none of the differences  $  \alpha - \alpha  ^  \prime  $,
-$  \beta - \beta  ^  \prime  $,
-$  \gamma - \gamma  ^  \prime  $
-are integers, then interchanging in (2) the positions of  $  \alpha $
-and  $  \alpha  ^  \prime  $
-or of  $  \gamma $
-and  $  \gamma  ^  \prime  $,
-four solutions of a Papperitz equation are obtained. In addition a Papperitz equation remains unchanged if the positions of the triples  $  ( \alpha , \alpha  ^  \prime  , a) $,
-$  ( \beta , \beta  ^  \prime  , b) $,
-$  ( \gamma , \gamma  ^  \prime  , c) $
-are rearranged; all these rearrangements provide 24 special solutions of a Papperitz equation (1), which were first obtained by E.E. Kummer [[#References|[5]]].
 ====References====
 <table><TR><TD valign="top">[1]</TD> <TD valign="top"> B. Riemann, "Beiträge zur Theorie der durch Gauss'sche Reihe <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113056.png" /> darstellbare Functionen" , ''Gesammelte math. Werke'' , Dover, reprint (1953) pp. 67–85</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> E. Papperitz, "Ueber verwandte <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113057.png" />-Functionen" ''Math. Ann.'' , '''25''' (1885) pp. 212–221 {{MR|1510304}} {{ZBL|}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> E.T. Whittaker, G.N. Watson, "A course of modern analysis" , Cambridge Univ. Press (1952) pp. Chapt. 6 {{MR|1424469}} {{MR|0595076}} {{MR|0178117}} {{MR|1519757}} {{ZBL|0951.30002}} {{ZBL|0108.26903}} {{ZBL|0105.26901}} {{ZBL|53.0180.04}} {{ZBL|47.0190.17}} {{ZBL|45.0433.02}} {{ZBL|33.0390.01}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> V.V. Golubev, "Vorlesungen über Differentialgleichungen im Komplexen" , Deutsch. Verlag Wissenschaft. (1958) (Translated from Russian) {{MR|0100119}} {{ZBL|}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> E.E. Kummer, "Ueber die hypergeometrische Reihe <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113058.png" />" ''J. Reine Angew. Math.'' , '''15''' (1836) pp. 39–83; 127–172</TD></TR></table>

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Difference between revisions of "Papperitz equation"

Revision as of 14:52, 7 June 2020

References