Description Usage Arguments Details Value Examples
Calculates Spherical Bessel functions from 0 to nmax.
1 | bess.sph(nmax, x, code = "C")
|
nmax |
The maximum order of j_n(x) |
x |
The argument of j_n(x) |
code |
If you prefer to use native R or C language. The algorithm is the same. |
bess.sph
calculates the Spherical Bessel
functions using downward recurrence, from j_nmax(x) to j_0(x).
The system of equations is given by S_n(x)=n/x,
r[n]=j_n/j_{n+1} and
c_n=j_n'(x)/j_n(x). The system can be solved by means of
the recurrence relations of the Spherical Bessel functions
r[n-1]+ 1/r[n]=S[2n+1]
nr[n-1]-(n+1)/r[n]=(2n+1)c[n]
that can be rewriten
r[n]=S[n+2]+c[n+1]
1/r[n]=S[n ]-c[n ].
The logarithmic derivatives obeys the relation,
(S[n+2]-c[n])(S[n]+C[n])=1.
The values can be calculated upward or downward.
An array of Spherical Bessel functions and its derivatives
from 0 to nmax
at point x
, and also the logarithmic
derivative c_n=j_n'/j_n and the ratio ρ_n=j_n/j_{n+1}.
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