Schläfli integral

An integral representation of the Bessel functions for any $ n $:

$$ \tag{* } J _ {n} ( z) = \frac{1} \pi \int\limits _ { 0 } ^ \pi \cos ( n \theta - z \sin \theta ) d \theta + $$

$$ - \frac{\sin n \pi } \pi \int\limits _ { 0 } ^ \infty e ^ {- n \theta - z \sinh \theta } d \theta , $$

when $ \mathop{\rm Re} z > 0 $. It is valid for all integer $ n $. Formula (*) can be derived from

$$ J _ {n} = \frac{z ^ {n} }{2 ^ {\pi + 1 } \pi i } \int\limits _ {- \infty } ^ { ( } 0+) t ^ {-} n- 1 \mathop{\rm exp} \left ( t - \frac{z ^ {2} }{4t} \right ) dt. $$

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

An integral representation of the Legendre polynomials:

$$ P _ {n} ( z) = \frac{1}{2 \pi i } \int\limits _ { C } \frac{( t ^ {2} - 1) ^ {n} }{2 ^ {n} ( t- z) ^ {n+} 1 } dt, $$

where $ C $ is a contour making one counter-clockwise turn around $ z $. 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 $ J _ {n} ( z) $ is valid for unrestricted values of $ n $( 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)