Spherical Bessel Function of the First Kind
The spherical Bessel function of the first kind, denoted j_nu(z), is defined by
where J_nu(z) is a Bessel function of the first kind and, in general, z and nu are complex numbers.
The function is most commonly encountered in the case nu=n an integer, in which case it is given by
Equation (4) shows the close connection between j_n(0) and the sinc function sinc(x)=sinx/x.
Spherical Bessel functions j_nu(z) are implemented in the Wolfram Language as SphericalBesselJ [nu, z] using the definition
which differs from the "traditional version" along the branch cut of the square root function, i.e., the negative real axis (e.g., at j_0(-1)), but has nicer analytic properties for complex z (Falloon 2001).
The first few functions are
which includes the special value
| j_0(z)=sinc(z). |
(9)
|
See also
Sinc Function, Spherical Bessel Differential Equation, Bessel Function of the Second Kind, Poisson Integral Representation, Rayleigh's Formulas, Spherical Bessel Function of the Second KindExplore with Wolfram|Alpha
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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.Falloon, P. E. Theory and Computation of Spheroidal Harmonics with General Arguments. Masters thesis. Perth, Australia: University of Western Australia, 2001. https://doi.org/10.26182/9pna-sh20.Referenced on Wolfram|Alpha
Spherical Bessel Function of the First KindCite this as:
Weisstein, Eric W. "Spherical Bessel Function of the First Kind." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SphericalBesselFunctionoftheFirstKind.html