scipy.special.iv#
- scipy.special.iv(v, z, out=None) = <ufunc 'iv'>#
- Modified Bessel function of the first kind of real order. - Parameters:
- varray_like
- Order. If z is of real type and negative, v must be integer valued. 
- zarray_like of float or complex
- Argument. 
- outndarray, optional
- Optional output array for the function values 
 
- Returns:
- scalar or ndarray
- Values of the modified Bessel function. 
 
 - See also - Notes - For real z and \(v \in [-50, 50]\), the evaluation is carried out using Temme’s method [1]. For larger orders, uniform asymptotic expansions are applied. - For complex z and positive v, the AMOS [2] zbesi routine is called. It uses a power series for small z, the asymptotic expansion for large abs(z), the Miller algorithm normalized by the Wronskian and a Neumann series for intermediate magnitudes, and the uniform asymptotic expansions for \(I_v(z)\) and \(J_v(z)\) for large orders. Backward recurrence is used to generate sequences or reduce orders when necessary. - The calculations above are done in the right half plane and continued into the left half plane by the formula, \[I_v(z \exp(\pm\imath\pi)) = \exp(\pm\pi v) I_v(z)\]- (valid when the real part of z is positive). For negative v, the formula \[I_{-v}(z) = I_v(z) + \frac{2}{\pi} \sin(\pi v) K_v(z)\]- is used, where \(K_v(z)\) is the modified Bessel function of the second kind, evaluated using the AMOS routine zbesk. - References [1]- Temme, Journal of Computational Physics, vol 21, 343 (1976) [2]- Donald E. Amos, “AMOS, A Portable Package for Bessel Functions of a Complex Argument and Nonnegative Order”, http://netlib.org/amos/ - Examples - Evaluate the function of order 0 at one point. - >>> from scipy.special import iv >>> iv(0, 1.) 1.2660658777520084 - Evaluate the function at one point for different orders. - >>> iv(0, 1.), iv(1, 1.), iv(1.5, 1.) (1.2660658777520084, 0.565159103992485, 0.2935253263474798) - The evaluation for different orders can be carried out in one call by providing a list or NumPy array as argument for the v parameter: - >>> iv([0, 1, 1.5], 1.) array([1.26606588, 0.5651591 , 0.29352533]) - Evaluate the function at several points for order 0 by providing an array for z. - >>> import numpy as np >>> points = np.array([-2., 0., 3.]) >>> iv(0, points) array([2.2795853 , 1. , 4.88079259]) - If z is an array, the order parameter v must be broadcastable to the correct shape if different orders shall be computed in one call. To calculate the orders 0 and 1 for an 1D array: - >>> orders = np.array([[0], [1]]) >>> orders.shape (2, 1) - >>> iv(orders, points) array([[ 2.2795853 , 1. , 4.88079259], [-1.59063685, 0. , 3.95337022]]) - Plot the functions of order 0 to 3 from -5 to 5. - >>> import matplotlib.pyplot as plt >>> fig, ax = plt.subplots() >>> x = np.linspace(-5., 5., 1000) >>> for i in range(4): ... ax.plot(x, iv(i, x), label=f'$I_{i!r}$') >>> ax.legend() >>> plt.show() 