scipy.special.ive#
- scipy.special.ive(v, z, out=None) = <ufunc 'ive'>#
- Exponentially scaled modified Bessel function of the first kind. - Defined as: - ive(v, z) = iv(v, z) * exp(-abs(z.real)) - For imaginary numbers without a real part, returns the unscaled Bessel function of the first kind - iv.- Parameters:
- varray_like of float
- Order. 
- zarray_like of float or complex
- Argument. 
- outndarray, optional
- Optional output array for the function values 
 
- Returns:
- scalar or ndarray
- Values of the exponentially scaled modified Bessel function. 
 
 - See also - Notes - For positive v, the AMOS [1] 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. - iveis useful for large arguments z: for these,- iveasily overflows, while- ivedoes not due to the exponential scaling.- References [1]- Donald E. Amos, “AMOS, A Portable Package for Bessel Functions of a Complex Argument and Nonnegative Order”, http://netlib.org/amos/ - Examples - In the following example - ivreturns infinity whereas- ivestill returns a finite number.- >>> from scipy.special import iv, ive >>> import numpy as np >>> import matplotlib.pyplot as plt >>> iv(3, 1000.), ive(3, 1000.) (inf, 0.01256056218254712) - Evaluate the function at one point for different orders by providing a list or NumPy array as argument for the v parameter: - >>> ive([0, 1, 1.5], 1.) array([0.46575961, 0.20791042, 0.10798193]) - Evaluate the function at several points for order 0 by providing an array for z. - >>> points = np.array([-2., 0., 3.]) >>> ive(0, points) array([0.30850832, 1. , 0.24300035]) - Evaluate the function at several points for different orders by providing arrays for both v for z. Both arrays have to be broadcastable to the correct shape. To calculate the orders 0, 1 and 2 for a 1D array of points: - >>> ive([[0], [1], [2]], points) array([[ 0.30850832, 1. , 0.24300035], [-0.21526929, 0. , 0.19682671], [ 0.09323903, 0. , 0.11178255]]) - Plot the functions of order 0 to 3 from -5 to 5. - >>> fig, ax = plt.subplots() >>> x = np.linspace(-5., 5., 1000) >>> for i in range(4): ... ax.plot(x, ive(i, x), label=fr'$I_{i!r}(z)\cdot e^{{-|z|}}$') >>> ax.legend() >>> ax.set_xlabel(r"$z$") >>> plt.show() 