Sonin integral

The representation of a cylinder function (cf. Cylinder functions) by a contour integral

$$ J _ \nu ( z) = \ \frac{1}{2 \pi i } \int\limits _ {- \infty } ^ { {( } 0+) } e ^ {z ( t ^ {2} - t ) /2t } t ^ {- \nu - 1 } dt, $$

where $ \nu $ is arbitrary and $ \mathop{\rm Re} z > 0 $ or $ - \pi /2 < \mathop{\rm arg} z < \pi /2 $. Integrals of this type were studied by N.Ya. Sonin (1870).

An integral of the form below is sometimes called a Sonin integral:

$$ J _ {m + n + 1 } ( x) = $$

$$ = \ \frac{x ^ {n + 1 } }{2 ^ {n} \Gamma ( n + 1) } \int\limits _ { 0 } ^ { \pi /2 } J _ {m} ( x \sin t) \sin ^ {m + 1 } t \cos ^ {2n + 1 } t dt, $$

$$ \mathop{\rm Re} m, \mathop{\rm Re} n > - 1. $$

References

[1]	M.A. Lavrent'ev, B.V. Shabat, "Methoden der komplexen Funktionentheorie" , Deutsch. Verlag Wissenschaft. (1967) (Translated from Russian)
[2]	E. Jahnke, F. Emde, F. Lösch, "Tafeln höheren Funktionen" , Teubner (1966)

Comments

In Western practice one usually writes Sonine integral. Transformed versions of the contour integral were independently obtained by L. Schläfli (1873) and integrals of this type are also called Schläfli integrals. The second mentioned integral is known as Sonine's first finite integral.

References

[a1]	G.N. Watson, "The theory of Bessel functions" , 1 , Cambridge Univ. Press (1944) pp. Formulas 6.2(2), 12.11(1)