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Sommerfeld's Formula

There are (at least) two equations known as Sommerfeld's formula. The first is

J_nu(z)=1/(2pi)int_(-eta+iinfty)^(2pi-eta+iinfty)e^(izcost)e^(inu(t-pi/2))dt,

where J_nu(z) is a Bessel function of the first kind. The second states that under appropriate restrictions,

int_0^inftyJ_0(taur)e^(-|x|sqrt(tau^2-k^2))(taudtau)/(sqrt(tau^2-k^2))=(e^(-i|r|sqrt(k^2+x^2)))/(sqrt(r^2+x^2)).

See also

Weyrich's Formula

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References

Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 1472 and 1474, 1980.

Referenced on Wolfram|Alpha

Sommerfeld's Formula

Cite this as:

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

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