Spherical Bessel Function of the First Kind
المؤلف:
Abramowitz, M. and Stegun, I. A
المصدر:
"Spherical Bessel Functions." §10.1 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover
الجزء والصفحة:
...
25-11-2018
937
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
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)
|
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.
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