Modified Bessel function of the first kind, order 0.

Usually denoted \(I_0\).

xarray_like of float

Argument of the Bessel function.

outndarray, shape = x.shape, dtype = float

The modified Bessel function evaluated at each of the elements of x.


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).



C. W. Clenshaw, “Chebyshev series for mathematical functions”, in National Physical Laboratory Mathematical Tables, vol. 5, London: Her Majesty’s Stationery Office, 1962.


M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, 10th printing, New York: Dover, 1964, pp. 379.


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