scipy.special.i1e#

scipy.special.i1e(x, out=None) = <ufunc 'i1e'>#

Exponentially scaled modified Bessel function of order 1.

Defined as:

i1e(x) = exp(-abs(x)) * i1(x)

Parameters:

xarray_like: Argument (float)
outndarray, optional: Optional output array for the function values

Returns:

Iscalar or ndarray: Value of the exponentially scaled modified Bessel function of order 1 at x.

See also

iv: Modified Bessel function of the first kind
i1: Modified Bessel function of order 1

Notes

The range is partitioned into the two intervals [0, 8] and (8, infinity). Chebyshev polynomial expansions are employed in each interval. The polynomial expansions used are the same as those in i1, but they are not multiplied by the dominant exponential factor.

This function is a wrapper for the Cephes [1] routine i1e. i1e is useful for large arguments x: for these, i1 quickly overflows.

Array API Standard Support

i1e has support for Python Array API Standard compatible backends in addition to NumPy. The following combinations of backend and device (or other capability) are supported.

Library	CPU	GPU
NumPy	✅	n/a
CuPy	n/a	✅
PyTorch	✅	✅
JAX	✅	✅
Dask	✅	n/a

For the NumPy backend, this function supports all NumPy ufunc keyword arguments. Other backends may support out, but none of the other ufunc kwargs. out is typically supported for CuPy and PyTorch, but not currently in cases where SciPy relies on a generic Array API implementation or, for PyTorch on CPU, falls back to the NumPy backend. out is never supported for JAX because JAX arrays are immutable.

See Support for the array API standard for more information.

References

[1]

Cephes Mathematical Functions Library, http://www.netlib.org/cephes/

Examples

In the following example i1 returns infinity whereas i1e still returns a finite number.

>>> from scipy.special import i1, i1e
>>> i1(1000.), i1e(1000.)
(inf, 0.01261093025692863)

Calculate the function at several points by providing a NumPy array or list for x:

>>> import numpy as np
>>> i1e(np.array([-2., 0., 6.]))
array([-0.21526929,  0.        ,  0.15205146])

Plot the function between -10 and 10.

>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> x = np.linspace(-10., 10., 1000)
>>> y = i1e(x)
>>> ax.plot(x, y)
>>> plt.show()