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Dixon-Ferrar Formula

Let J_nu(z) be a Bessel function of the first kind, Y_nu(z) a Bessel function of the second kind, and K_nu(z) a modified Bessel function of the first kind. Also let R[z]>0 and -1/2<R[nu]<1/2. Then the Dixon-Ferrar formula

int_0^inftyK_(2nu)(2zsinht)dt=(8cos(nupi))/(pi^2)[J_nu^2(z)+Y_nu^2(z)]

(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[(nu-mu)pi])[J_nu(z)Y_mu(z)-J_mu(z)Y_nu(z)]

(Magnus and Oberhettinger 1948, p. 44; Gradshteyn and Ryzhik 2000, p. 703) for R[z]>1 and -1<R[nu-mu]<1.

See also

Nicholson's Formula, Watson's Formula

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References

Gradshteyn, I. S. and Ryzhik, I. M. Eqns. 6.518 and 6.649.1 in Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, pp. 657 and 703, 2000.Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1476, 1980.Magnus, W. and Oberhettinger, F. Formeln und Sätze für die speziellen Funktionen der mathematischen Physik, 2nd ed. Berlin: Springer-Verlag, 1948.

Referenced on Wolfram|Alpha

Dixon-Ferrar Formula

Cite this as:

Weisstein, Eric W. "Dixon-Ferrar Formula." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Dixon-FerrarFormula.html

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