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:

${\displaystyle \int _{0}^{\infty }J_{z}(at)J_{z}(bt)J_{z}(ct)t^{1-z}\,dt={\frac {2^{z-1}\Delta (a,b,c)^{2z-1}}{\pi ^{1/2}\Gamma (z+{\tfrac {1}{2}})(abc)^{z}}}}$

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, JSTOR 2046994, MR 0962812