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Date: 1-10-2019
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A modified spherical Bessel function of the second kind, also called a "spherical modified Bessel function of the first 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
(1) |
where is a modified Bessel function of the second kind (Arfken 1985, p. 633)
For positive , the first few values for small nonnegative integer indices are
(2) |
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(3) |
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(4) |
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(5) |
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(6) |
(OEIS A001498).
Writing
(7) |
the are given by the recurrence equation
(8) |
together with
(9) |
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(10) |
(Abramowitz and Stegun 1972, p. 444).
has no definite parity (Arfken 1985, p. 633).
is related to the spherical Hankel function of the first kind by
(11) |
for and integer (Arfken 1985, p. 633).
They also satisfy the differential identities
(12) |
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(13) |
and the recurrence relations
(14) |
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(15) |
(Arfken 1985, p. 634).
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. 663-634, 1985.
Sloane, N. J. A. Sequence A001498 in "The On-Line Encyclopedia of Integer Sequences."
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