spherical_in#
- scipy.special.spherical_in(n, z, derivative=False)[source]#
- Modified spherical Bessel function of the first kind or its derivative. - Defined as [1], \[i_n(z) = \sqrt{\frac{\pi}{2z}} I_{n + 1/2}(z),\]- where \(I_n\) is the modified Bessel function of the first kind. - Parameters:
- nint, array_like
- Order of the Bessel function (n >= 0). 
- zcomplex or float, array_like
- Argument of the Bessel function. 
- derivativebool, optional
- If True, the value of the derivative (rather than the function itself) is returned. 
 
- Returns:
- inndarray
 
 - Notes - 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 [2], \[ \begin{align}\begin{aligned}i_n' = i_{n-1} - \frac{n + 1}{z} i_n.\\i_1' = i_0\end{aligned}\end{align} \]- Added in version 0.18.0. - References [AS]- Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. - Examples - 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 \([1, 2]\): - >>> import numpy as np >>> 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 >>> 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() 