numpy.i0#

numpy.i0(x)[source]#

Modified Bessel function of the first kind, order 0.

Usually denoted \(I_0\).

Parameters

xarray_like of float: Argument of the Bessel function.

Returns

outndarray, shape = x.shape, dtype = float: The modified Bessel function evaluated at each of the elements of x.

See also

scipy.special.i0, scipy.special.iv, scipy.special.ive

Notes

The scipy implementation is recommended over this function: it is a proper ufunc written in C, and more than an order of magnitude faster.

We use the algorithm published by Clenshaw [1] and referenced by Abramowitz and Stegun [2], for which the function domain is partitioned into the two intervals [0,8] and (8,inf), and Chebyshev polynomial expansions are employed in each interval. Relative error on the domain [0,30] using IEEE arithmetic is documented [3] as having a peak of 5.8e-16 with an rms of 1.4e-16 (n = 30000).

References

1: C. W. Clenshaw, “Chebyshev series for mathematical functions”, in National Physical Laboratory Mathematical Tables, vol. 5, London: Her Majesty’s Stationery Office, 1962.
2: M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, 10th printing, New York: Dover, 1964, pp. 379. https://personal.math.ubc.ca/~cbm/aands/page_379.htm
3: https://metacpan.org/pod/distribution/Math-Cephes/lib/Math/Cephes.pod#i0:-Modified-Bessel-function-of-order-zero

Examples

>>> np.i0(0.)
array(1.0)
>>> np.i0([0, 1, 2, 3])
array([1.        , 1.26606588, 2.2795853 , 4.88079259])

numpy.logaddexp2

numpy.sinc