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Mehler's Bessel Function Formula

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For x>0,

J_0(x)=2/piint_0^inftysin(xcosht)dt
(1)
Y_0(x)=-2/piint_0^inftycos(xcosht)dt,
(2)

where J_0(x) is a zeroth order Bessel function of the first kind and Y_0(x) is a zeroth order Bessel function of the second kind.

See also

Bessel Function of the First Kind, Bessel Function of the Second Kind

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References

Itô, K. (Ed.). Encyclopedic Dictionary of Mathematics, 2nd ed., Vol. 2. Cambridge, MA: MIT Press, p. 1802, 1986.Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1472, 1980.

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Mehler's Bessel Function Formula

Cite this as:

Weisstein, Eric W. "Mehler's Bessel Function Formula." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MehlersBesselFunctionFormula.html

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