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
where I_n(z) is a modified Bessel function of the first kind (Arfken 1985, p. 633).
For positive x, the first few values for small nonnegative integer indices are
Writing
| i_n(z)=g_n(z)sinhz+g_(-(n+1))(z)coshz, |
(7)
|
the g_n are given by the recurrence equation
| g_(n-1)(z)-g_(n+1)(z)=(2n+1)z^(-1)g_n(z) |
(8)
|
together with
(Abramowitz and Stegun 1972, p. 443).
The parity of i_n(x) is (-1)^n (Arfken 1985, p. 633).
i_n(x) is related to the spherical Bessel function of the first kind j_n(x) by
| i_n(x)=i^(-n)j_n(ix) |
(11)
|
for x>0 and integer n (Arfken 1985, p. 633).
They also satisfy the differential identities
and the recurrence relations
(Arfken 1985, p. 634).
See also
Modified Bessel Function of the First Kind, Modified Spherical Bessel Function of the Second KindExplore with Wolfram|Alpha
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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.Sloane, N. J. A. Sequences A094674 and A094675 in "The On-Line Encyclopedia of Integer Sequences."Referenced on Wolfram|Alpha
Modified Spherical Bessel Function of the First KindCite this as:
Weisstein, Eric W. "Modified Spherical Bessel Function of the First Kind." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ModifiedSphericalBesselFunctionoftheFirstKind.html