The Dixon-Ferrar formula is an integral identity involving Bessel functions. Let 
be a Bessel function of the first kind,
a Bessel function of the second kind,
and 
a modified Bessel function
of the second kind. For ![Re[z]>0](/images/equations/Dixon-FerrarFormula/Inline4.svg)
and ![|Re[nu]|<1/2](/images/equations/Dixon-FerrarFormula/Inline5.svg)
, the Dixon-Ferrar formula gives
![int_0^inftyK_(2nu)(2zsinht)dt=(pi^2)/(8cos(nupi))[J_nu^2(z)+Y_nu^2(z)]](/images/equations/Dixon-FerrarFormula/NumberedEquation1.svg) |
(Magnus and Oberhettinger 1948, p. 45; Iyanaga and Kawada 1980, p. 1476; Gradshteyn and Ryzhik 2000, p. 657).
This is a special case of the more general formula
![int_0^inftyK_(mu-nu)(2zsinht)e^((mu+nu)t)dt
=(pi^2)/(4sin[(mu-nu)pi])[J_(-nu)(z)Y_(-mu)(z)-J_(-mu)(z)Y_(-nu)(z)]](/images/equations/Dixon-FerrarFormula/NumberedEquation2.svg) |
(Magnus and Oberhettinger 1948, p. 44; Gradshteyn and Ryzhik 2000, p. 703) for ![Re[z]>0](/images/equations/Dixon-FerrarFormula/Inline6.svg)
and ![|Re[mu-nu]|<1](/images/equations/Dixon-FerrarFormula/Inline7.svg)
.
At 
,
the quotient is interpreted by continuity.
