Numerical implementation of logarithmic integral function

[Qazalbash]

Hi, it would be nice to have a Logarithmic integral function¹. It is defined as,

$$ \displaystyle\mathrm{li}(x)=\int_{0}^{x}\frac{\mathrm{d}{t}}{\ln{(t)}} $$

Footnotes

1. https://en.wikipedia.org/wiki/Logarithmic_integral_function ↩

[jakevdp]

Hi - thanks for the request! Agreed this would be useful. It's interesting that I couldn't find any function for this (or any feature request) for scipy.special, but maybe I just wasn't searching for the right keywords.

Implementing this in JAX would probably involve finding an appropriate series expansion for the function, which may be somewhat tricky given its singularity at x=1. I'm going to mark this as a P3 enhancement for now.

[carlosgmartin]

Related: Efficiently calculating the logarithmic integral with complex argument.

[jakevdp]

JAX does implement the exponential integral expi.

It looks like you could comput li using that with a change of variables