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Nicholson's Formula

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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. Then

J_nu^2(z)+Y_nu^2(z)=8/(pi^2)int_0^inftyK_0(2zsinht)cosh(2nut)dt

for R[z]>0, where R[z] denotes the real part of z.

See also

Bessel Function of the First Kind, Bessel Function of the Second Kind, Dixon-Ferrar Formula, Modified Bessel Function of the First Kind, Watson's Formula

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References

Gradshteyn, I. S. and Ryzhik, I. M. Eqn. 6.664.4 in Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 707, 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, p. 44, 1948.

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Nicholson's Formula

Cite this as:

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

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