Spherical Bessel Function of the First Kind

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

			(2)
			(3)
			(4)

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

			(6)
			(7)
			(8)

which includes the special value

(9)