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Difference between revisions of "Papperitz equation"

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====References====
 
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<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</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</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)</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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<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>

Revision as of 17:00, 15 April 2012

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=24524
This article was adapted from an original article by M.V. Fedoryuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article