Schläfli integral

An integral representation of the Bessel functions for any :

(*)

when . It is valid for all integer . Formula (*) can be derived from

Formula (*) was first given by L. Schläfli .

An integral representation of the Legendre polynomials:

where is a contour making one counter-clockwise turn around . This representation was first given by L. Schläfli [2].

References

[1]	L. Schläfli, "Eine Bemerkung zu Herrn Neumanns Untersuchungen über die Besselschen Funktionen" Math. Ann. , 3 : 1 (1871) pp. 134–149
[2]	L. Schläfli, "Über die zwei Heine'schen Kugelfunktionen mit beliebigem Parameter und ihre ausnahmslose Darstellung durch bestimmte Integrale" , H. Koerber , Berlin (1881)
[3]	E.T. Whittaker, G.N. Watson, "A course of modern analysis" , Cambridge Univ. Press (1952) pp. Chapt. 6
[4]	A. Krazer, W. Franz, "Transzendente Funktionen" , Akademie Verlag (1960)

Comments

The reduction of the Schläfli integral to the second integral representation for is valid for unrestricted values of (see also [a3], 6.2

and ). The integral representation for the Legendre polynomials follows from the Rodrigues formula, similarly as for the Jacobi polynomials (cf. [a2], (4.4.6) and (4.8.1)).

References

[a1]	F.W.J. Olver, "Asymptotics and special functions" , Acad. Press (1974)
[a2]	G. Szegö, "Orthogonal polynomials" , Amer. Math. Soc. (1975)
[a3]	G.N. Watson, "A treatise on the theory of Bessel functions" , 1 , Cambridge Univ. Press (1952)