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Kneser-Sommerfeld Formula


Let J_nu(z) be a Bessel function of the first kind, N_nu(z) a Bessel function of the second kind, and j_(nu,n)(z) the zeros of z^(-nu)J_nu(z) in order of ascending real part. Then for 0<x<X<1 and R[z]>0,

 (piJ_nu(xz))/(4J_nu(z))[J_nu(z)N_nu(Xz)-N_nu(z)J_nu(Xz)]=sum_(n=1)^infty(J_nu(j_(nu,n)x)J_nu(j_(nu,n)X))/((z^2-j_(nu,n)^2)J_(nu,n)^('2)(j_(nu,n))).

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References

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

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Kneser-Sommerfeld Formula

Cite this as:

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

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