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Modified Spherical Bessel Function of the Second Kind

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ModifiedSphericalBesselK

A modified spherical Bessel function of the second kind, also called a "spherical modified Bessel function of the second kind" (Arfken 1985) or (regrettably) a "modified spherical Bessel function of the third kind" (Abramowitz and Stegun 1972, p. 443), is the second solution to the modified spherical Bessel differential equation, given by

k_n(x)=sqrt(2/(pix))K_(n+1/2)(x),
(1)

where K_n(z) is a modified Bessel function of the second kind (Arfken 1985, p. 633).

For nonnegative integers n, the NIST Digital Library of Mathematical Functions uses a different normalization, namely k_n^(DLMF)(x)=sqrt(pi/(2x))K_(n+1/2)(x)=(pi/2)k_n(x) (NIST, eq. 10.47.9).

For positive x, the first few values for small nonnegative integer indices are

k_0(x)=(e^(-x))/x
(2)
k_1(x)=(e^(-x)(x+1))/(x^2)
(3)
k_2(x)=(e^(-x)(x^2+3x+3))/(x^3)
(4)
k_3(x)=(e^(-x)(x^3+6x^2+15x+15))/(x^4)
(5)
k_4(x)=(e^(-x)(x^4+10x^3+45x^2+105x+105))/(x^5)
(6)

(OEIS A001498).

Writing

k_n(z)=e^(-z)f_n(z),
(7)

the f_n for n>=2 are given by the recurrence equation

f_n(z)=f_(n-2)(z)+(2n-1)z^(-1)f_(n-1)(z)
(8)

together with

f_0(z)=z^(-1)
(9)
f_1(z)=(z+1)/(z^2)
(10)

(Abramowitz and Stegun 1972, p. 444).

k_n(x) has no definite parity (Arfken 1985, p. 633).

k_n(x) is related to the spherical Hankel function of the first kind h_n^((1))(x) by

k_n(x)=-i^nh_n^((1))(ix)
(11)

for x>0 and nonnegative integer n (Arfken 1985, p. 633).

For x>0 and nonnegative integers n, they also satisfy the differential identities

k_(n+1)(x)=-x^nd/(dx)(x^(-n)k_n)
(12)
k_n(x)=(-1)^nx^n(d/(xdx))^n(e^(-x))/x,
(13)

and, for n>=1, the recurrence relations

k_(n-1)(x)-k_(n+1)(x)=-(2n+1)/xk_n(x)
(14)
nk_(n-1)(x)+(n+1)k_(n+1)(x)=-(2n+1)k_n^'(x)
(15)

(Arfken 1985, p. 634).

See also

Bessel Polynomial, Modified Bessel Function of the Second Kind, Modified Spherical Bessel Function of the First Kind

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References

Abramowitz, M. and Stegun, I. A. (Eds.). "Modified Spherical Bessel Functions." §10.2 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 443-445, 1972.Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 633-634, 1985.NIST Digital Library of Mathematical Functions. "Definitions and Basic Properties." §10.47(ii), eq. 10.47.9. https://dlmf.nist.gov/10.47.E9.Sloane, N. J. A. Sequence A001498 in "The On-Line Encyclopedia of Integer Sequences."

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Modified Spherical Bessel Function of the Second Kind

Cite this as:

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

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