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Watson's Formula

Watson's formula is an integral identity for Bessel functions. 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 Re[z]>0 and require Re[mu-nu]<1. Then

J_mu(z)Y_nu(z)-J_nu(z)Y_mu(z)=(4sin[(mu-nu)pi])/(pi^2)int_0^inftyK_(nu-mu)(2zsinht)e^(-(mu+nu)t)dt.

The fourth edition of Gradshteyn and Ryzhik (2000), Iyanaga and Kawada (1980), and Ito (1987) erroneously give the exponential with a plus sign. A related integral is given by

J_nu(z)(partialY_nu(z))/(partialnu)-Y_nu(z)(partialJ_nu(z))/(partialnu)=-4/piint_0^inftyK_0(2zsinht)e^(-2nut)dt

for Re[z]>0.

See also

Dixon-Ferrar Formula, Nicholson's Formula, Watson-Nicholson Formula, Watson's Identities, Watson's Triple Integrals

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References

Gradshteyn, I. S. and Ryzhik, I. M. Eqns. 6.617.1 and 6.617.2 in Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 710, 2000.Itô, K. (Ed.). Encyclopedic Dictionary of Mathematics, 2nd ed. Cambridge, MA: MIT Press, p. 1806, 1987.Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1476, 1980.

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Watson's Formula

Cite this as:

Weisstein, Eric W. "Watson's Formula." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/WatsonsFormula.html

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