The spherical Bessel function of the first kind, denoted , is defined by

(1)

where is a Bessel
function of the first kind and, in general, and are complex numbers.

The function is most commonly encountered in the case an integer, in which case it is given by

Equation (4 ) shows the close connection between and the sinc function .

Spherical Bessel functions are implemented in the Wolfram
Language as SphericalBesselJ [nu ,
z ] using the definition

(5)

which differs from the "traditional version" along the branch cut of the square root function, i.e., the negative
real axis (e.g., at ), but has nicer analytic properties for complex
(Falloon 2001).

The first few functions are

which includes the special value

(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 Kind
Explore with Wolfram|Alpha
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. http://www.physics.uwa.edu.au/pub/Theses/2002/Falloon/Masters_Thesis.pdf . Referenced
on Wolfram|Alpha Spherical Bessel
Function of the First Kind
Cite this as:
Weisstein, Eric W. "Spherical Bessel Function of the First Kind." From MathWorld --A Wolfram Web Resource.
https://mathworld.wolfram.com/SphericalBesselFunctionoftheFirstKind.html

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