jax.scipy.special.fresnel

jax.scipy.special.fresnel = <jax._src.custom_derivatives.custom_jvp object>

The Fresnel integrals.

JAX implementation of scipy.special.fresnel.

The Fresnel integrals are defined by

\[\begin{split}\begin{aligned} S(x) &= \int_0^x \sin\left(\pi t^2 / 2\right)\mathrm{d}t, \\ C(x) &= \int_0^x \cos\left(\pi t^2 / 2\right)\mathrm{d}t. \end{aligned}\end{split}\]

Parameters:

x (ArrayLike) – arraylike, real-valued.

Returns:

A tuple (S, C) of arrays containing the values of the Fresnel integrals.

Return type:

tuple[Array, Array]

Notes

The JAX version only supports real-valued inputs, and is based on the SciPy C++ implementation, see here. For float32 dtypes, the implementation is directly based on the Cephes implementation fresnlf.

As for the original Cephes implementation, the accuracy is only guaranteed in the domain [-10, 10]. Outside of that domain, one could observe divergence between the theoretical derivatives and the custom JVP implementation, especially for large input values.

Finally, for half-precision data types, float16 and bfloat16, the array elements are upcasted to float32 as the Cephes coefficients used in series expansions would otherwise lead to poor results. Other data types, like float8, are not supported.