Mathematical formula involving Bessel functions
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:
∫
0
∞
J
z
(
a
t
)
J
z
(
b
t
)
J
z
(
c
t
)
t
1
−
z
d
t
=
2
z
−
1
Δ
(
a
,
b
,
c
)
2
z
−
1
π
1
/
2
Γ
(
z
+
1
2
)
(
a
b
c
)
z
{\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
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