Modified Bessel function of the first kind, order 0.

Usually denoted \(I_0\). This function does broadcast, but will not “up-cast” int dtype arguments unless accompanied by at least one float or complex dtype argument (see Raises below).


x (array_like, dtype float or complex) – Argument of the Bessel function.


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

Return type

ndarray, shape = x.shape, dtype = x.dtype


TypeError – array cannot be safely cast to required type: If argument consists exclusively of int dtypes.

See also

scipy.special.i0(), scipy.special.iv(), scipy.special.ive()


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.

3 #i0:-Modified-Bessel-function-of-order-zero


>>> np.i0(0.)
array(1.0)  # may vary
>>> np.i0([0., 1. + 2j])
array([ 1.00000000+0.j        ,  0.18785373+0.64616944j])  # may vary