Modified Spherical Bessel Function of the First Kind
المؤلف:
Abramowitz, M. and Stegun, I. A.
المصدر:
"Modified Spherical Bessel Functions." §10.2 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover
الجزء والصفحة:
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25-3-2019
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Modified Spherical Bessel Function of the First Kind
A modified spherical Bessel function of the first kind (Abramowitz and Stegun 1972), also called a "spherical modified Bessel function of the first kind" (Arfken 1985), is the first solution to the modified spherical Bessel differential equation, given by
 |
(1)
|
where 
is a modified Bessel function of the first kind (Arfken 1985, p. 633).
For positive 
, the first few values for small nonnegative integer indices are
(OEIS A094674 and A094675).
Writing
 |
(7)
|
the 
are given by the recurrence equation
 |
(8)
|
together with
(Abramowitz and Stegun 1972, p. 443).
The parity of 
is 
(Arfken 1985, p. 633).
is related to the spherical Bessel function of the first kind 
by
 |
(11)
|
for 
and integer 
(Arfken 1985, p. 633).
They also satisfy the differential identities
and the recurrence relations
(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. 633-634, 1985.
Sloane, N. J. A. Sequences A094674 and A094675 in "The On-Line Encyclopedia of Integer Sequences."
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