`scipy.special._basic:riccati_yn`

source: build-install/usr/lib/python3.14/site-packages/scipy/special/_basic.py:1366

Signature

def riccati_yn(n, x)

Summary

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

Extended Summary

The Riccati-Bessel function of the second kind is defined here as $+ x y_{n} (x)$ , where $y_{n}$ is the spherical Bessel function of the second kind of order $n$ . Note that this is in contrast to a common convention that includes a minus sign in the definition.

This function computes the value and first derivative of the 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

yn : ndarray: Value of y0(x), ..., yn(x)
ynp : ndarray: First derivative y0'(x), ..., yn'(x)

Notes

The computation is carried out via ascending 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 second 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_yn
n = 5
z = 1.2
chi_n, chi_n_p = riccati_yn(n, z)

✓

chi_n
chi_n_p

✗

array([   0.93204,    0.66598,    3.30375,   22.5533 ,  177.40482,
       1659.57387])

chi_n_p[5]/chi_n[5]

✓

np.float64(-4.030059767479337)

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_yn
yn = spherical_yn(n, z)
ynp = spherical_yn(n, z, derivative=True)
ynp/yn + 1/z

✓

np.float64(-4.030059767479337)

Aliases

scipy.special.riccati_yn