spherical_yn(n, z, derivative=False)
Defined as ,
$$y_n(z) = \sqrt{\frac{\pi}{2z}} Y_{n + 1/2}(z),$$where $Y_n$ is the Bessel function of the second kind.
For real arguments, the function is computed using the ascending recurrence . For complex arguments, the definitional relation to the cylindrical Bessel function of the second kind is used.
The derivative is computed using the relations ,
$$y_n' = y_{n-1} - \frac{n + 1}{z} y_n. y_0' = -y_1$$Order of the Bessel function (n >= 0).
Argument of the Bessel function.
If True, the value of the derivative (rather than the function itself) is returned.
Spherical Bessel function of the second kind or its derivative.
The spherical Bessel functions of the second kind $y_n$ accept both real and complex second argument. They can return a complex type:
>>> from scipy.special import spherical_yn
... spherical_yn(0, 3+5j) (8.022343088587197-9.880052589376795j)
>>> type(spherical_yn(0, 3+5j)) <class 'numpy.complex128'>
We can verify the relation for the derivative from the Notes for $n=3$ in the interval $[1, 2]$ :
>>> from scipy.special import spherical_yn
... x = np.arange(1.0, 2.0, 0.01)
... np.allclose(spherical_yn(3, x, True),
... spherical_yn(2, x) - 4/x * spherical_yn(3, x)) True
The first few $y_n$ with real argument:
>>> import matplotlib.pyplot as plt
... from scipy.special import spherical_yn
... x = np.arange(0.0, 10.0, 0.01)
... fig, ax = plt.subplots()
... ax.set_ylim(-2.0, 1.0)
... ax.set_title(r'Spherical Bessel functions $y_n$')
... for n in np.arange(0, 4):
... ax.plot(x, spherical_yn(n, x), label=rf'$y_{n}$')
... plt.legend(loc='best')
... plt.show()
The following pages refer to to this document either explicitly or contain code examples using this.
scipy.special._spherical_bessel.spherical_yn
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