Sonine formula

In mathematics, Sonine's formula is any of several formulas involving Bessel functions found by Nikolay Yakovlevich Sonin.

One such formula is the following integral formula involving a product of three Bessel functions:

$\int_0^\infty J_z(at) J_z(bt)J_z(ct) t^{1-z}\,dt = \frac{2^{z-1}\Delta(a,b,c)^{2z-1}}{\pi^{1/2}\Gamma(z+\tfrac 12)(abc)^z}$

where Δ is the area of a triangle with given sides.

References

• Stempak, Krzysztof (1988), "A new proof of Sonine's formula", Proceedings of the American Mathematical Society 104 (2): 453–457, doi:10.2307/2046994, ISSN 0002-9939, MR 962812