scipy.special.

riccati_jn

scipy.special.riccati_jn(n, x)

Compute Riccati-Bessel function of the first kind and its derivative.

The Riccati-Bessel function of the first kind is defined as \(x j_n(x)\), where \(j_n\) is the spherical Bessel function of the first kind of order \(n\).

This function computes the value and first derivative of the Riccati-Bessel function for all orders up to and including n.

Parameters:

nint

Maximum order of function to compute

xfloat

Argument at which to evaluate

Returns:

jnndarray

Value of j0(x), …, jn(x)

jnpndarray

First derivative j0’(x), …, jn’(x)

Notes

The computation is carried out via backward recurrence, using the relation DLMF 10.51.1 [2].

Wrapper for a Fortran routine created by Shanjie Zhang and Jianming Jin [1].

References

[1]

Zhang, Shanjie and Jin, Jianming. “Computation of Special Functions”, John Wiley and Sons, 1996. https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html

[2]

NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/10.51.E1

Examples

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In practical applications, frequently the logarithmic derivative of the Riccati-Bessel functions is needed. We determine the logarithmic derivative of the Riccati-Bessel function of the first kind for order 5 and argument 1.2. The logarithmic derivative is obtained by dividing the derivative of the Riccati-Bessel function by the Riccati-Bessel function itself.

>>> from scipy.special import riccati_jn
>>> n = 5
>>> z = 1.2
>>> psi_n, psi_n_p = riccati_jn(n, z)
>>> psi_n
array([9.32039086e-01, 4.14341484e-01, 1.03814624e-01, 1.82194479e-02,
       2.46548893e-03, 2.71719094e-04])
>>> psi_n_p
array([0.36235775, 0.58675452, 0.24131711, 0.058266  , 0.01000115,
       0.00133333])
>>> psi_n_p[5]/psi_n[5]
np.float64(4.9070016327063115)

Alternatively, the logarithmic derivative of the Riccati-Bessel functions could be obtained from the corresponding spherical Bessel functions by making use of the definition of the Riccati-Bessel function in terms of the spherical Bessel function as given above.

>>> from scipy.special import spherical_jn
>>> jn = spherical_jn(n, z)
>>> jnp = spherical_jn(n, z, derivative=True)
>>> jnp/jn + 1/z
np.float64(4.907001632706311)

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