spherical_in(n, z, derivative=False)
Defined as ,
$$i_n(z) = \sqrt{\frac{\pi}{2z}} I_{n + 1/2}(z),$$where $I_n$ is the modified Bessel function of the first kind.
The function is computed using its definitional relation to the modified cylindrical Bessel function of the first kind.
The derivative is computed using the relations ,
$$i_n' = i_{n-1} - \frac{n + 1}{z} i_n. i_1' = i_0$$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.
Modified spherical Bessel function of the first kind or its derivative.
The modified spherical Bessel functions of the first kind $i_n$ accept both real and complex second argument. They can return a complex type:
>>> from scipy.special import spherical_in
... spherical_in(0, 3+5j) (-1.1689867793369182-1.2697305267234222j)
>>> type(spherical_in(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_in
... x = np.arange(1.0, 2.0, 0.01)
... np.allclose(spherical_in(3, x, True),
... spherical_in(2, x) - 4/x * spherical_in(3, x)) True
The first few $i_n$ with real argument:
>>> import matplotlib.pyplot as plt
... from scipy.special import spherical_in
... x = np.arange(0.0, 6.0, 0.01)
... fig, ax = plt.subplots()
... ax.set_ylim(-0.5, 5.0)
... ax.set_title(r'Modified spherical Bessel functions $i_n$')
... for n in np.arange(0, 4):
... ax.plot(x, spherical_in(n, x), label=rf'$i_{n}$')
... 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_in
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