Watson's Formula

Let  be a Bessel function of the first kind,  a Bessel function of the second kind, and  a modified Bessel function of the first kind. Also let  and require . Then

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

for .

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.