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Spherical Bessel Function of the First Kind


SphericalBesselj

The spherical Bessel function of the first kind, denoted j_nu(z), is defined by

 j_nu(z)=sqrt(pi/(2z))J_(nu+1/2)(z),
(1)

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

j_n(z)=2^nz^nsum_(k=0)^(infty)((-1)^k(k+n)!)/(k!(2k+2n+1)!)z^(2k)
(2)
=z^nsum_(k=0)^(infty)((-1)^k)/(k!(2k+2n+1)!!)((z^2)/2)^k
(3)
=(-1)^nz^n(d/(zdz))^n(sinz)/z.
(4)

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

 j_nu(z)=sqrt(pi/2)1/(sqrt(z))J_(nu+1/2)(z),
(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 j_0(-1)), but has nicer analytic properties for complex z (Falloon 2001).

The first few functions are

j_0(z)=(sinz)/z
(6)
j_1(z)=(sinz)/(z^2)-(cosz)/z
(7)
j_2(z)=(3/(z^3)-1/z)sinz-3/(z^2)cosz,
(8)

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 Kind

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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. 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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