spherical_jn(n, z, derivative=False)
Defined as ,
$$j_n(z) = \sqrt{\frac{\pi}{2z}} J_{n + 1/2}(z),$$where $J_n$ is the Bessel function of the first kind.
For real arguments greater than the order, the function is computed using the ascending recurrence . For small real or complex arguments, the definitional relation to the cylindrical Bessel function of the first kind is used.
The derivative is computed using the relations ,
$$j_n'(z) = j_{n-1}(z) - \frac{n + 1}{z} j_n(z). j_0'(z) = -j_1(z)$$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 first kind or its derivative.
The spherical Bessel functions of the first kind $j_n$ accept both real and complex second argument. They can return a complex type:
>>> from scipy.special import spherical_jn
... spherical_jn(0, 3+5j) (-9.878987731663194-8.021894345786002j)
>>> type(spherical_jn(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_jn
... x = np.arange(1.0, 2.0, 0.01)
... np.allclose(spherical_jn(3, x, True),
... spherical_jn(2, x) - 4/x * spherical_jn(3, x)) True
The first few $j_n$ with real argument:
>>> import matplotlib.pyplot as plt
... from scipy.special import spherical_jn
... x = np.arange(0.0, 10.0, 0.01)
... fig, ax = plt.subplots()
... ax.set_ylim(-0.5, 1.5)
... ax.set_title(r'Spherical Bessel functions $j_n$')
... for n in np.arange(0, 4):
... ax.plot(x, spherical_jn(n, x), label=rf'$j_{n}$')
... 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_jn
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