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


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

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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.

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.

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