`scipy.special._basic:riccati_jn`

source: build-install/usr/lib/python3.14/site-packages/scipy/special/_basic.py:1278

Signature

def riccati_jn(n, x)

Summary

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

Extended Summary

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

n : int: Maximum order of function to compute
x : float: Argument at which to evaluate

Returns

jn : ndarray: Value of j0(x), ..., jn(x)
jnp : ndarray: 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].

Examples

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
psi_n_p

✗

array([0.36236, 0.58675, 0.24132, 0.05827, 0.01   , 0.00133])

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)

Aliases

scipy.special.riccati_jn