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Spherical Bessel Function of the Second Kind

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SphericalBessely

The spherical Bessel function of the second kind, denoted y_nu(z) or n_nu(z), is defined by

y_nu(z)=sqrt(pi/(2z))Y_(nu+1/2)(z),
(1)

where Y_nu(z) is a Bessel function of the second kind and, in general, z and nu are complex numbers.

The spherical Bessel function of the second kind is implemented in the Wolfram Language as SphericalBesselY[n, z].

The function is most commonly encountered in the case nu=n an integer, in which case it is given by

y_n(z)=((-1)^(n+1))/(2^nz^(n+1))sum_(k=0)^(infty)((-1)^k(k-n)!)/(k!(2k-2n)!)z^(2k)
(2)
=((-1)^(n+1)sqrt(pi))/(2^nz^(n+1))sum_(k=0)^(infty)((-1)^k4^(n-k))/(Gamma(k+1)Gamma(1/2-n+k))z^(2k)
(3)
=((-1)^n)/(z^(n+1))((-1)^k)/(k!(2k-2n+1)!!)((z^2)/2)^k
(4)
=(-1)^(n+1)sqrt(pi/(2z))J_(-n-1/2)(z),
(5)

where J_n(z) is a Bessel function of the first kind.

Specific cases for small nonnegative n are given by

y_0(z)=-(cosz)/z
(6)
y_1(z)=-(cosz)/(z^2)-(sinz)/z
(7)
y_2(z)=-(3/(z^3)-1/z)cosz-3/(z^2)sinz.
(8)

See also

Spherical Bessel Differential Equation, Bessel Function of the Second Kind, Rayleigh's Formulas, Spherical Bessel Function of the First Kind

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References

Abramowitz, M. and Stegun, I. A. (Eds.). "Spherical Bessel Functions." §10.1 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 437-442, 1972.Arfken, G. "Spherical Bessel Functions." §11.7 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 622-636, 1985.

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Spherical Bessel Function of the Second Kind

Cite this as:

Weisstein, Eric W. "Spherical Bessel Function of the Second Kind." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SphericalBesselFunctionoftheSecondKind.html

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