JuliaMath / JuliaMath/SpecialFunctions.jl

exploit new expint(n,z) function

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#244 3 comments 2 reactions 0 assignees View on GitHub
enhancement
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Description

Might be good to define a few additional functions in terms of the new `expint` functions:

- [x] the `Ei` function (probably call it `expinti`) (#257)
- [ ] the `sinint` and `cosint` functions for complex arguments
- [x] incomplete gamma functions for complex arguments: #271
- [ ] the logarithmic integral function `li(x) = Ei(log(x))` (too trivial? or does it do something interesting for complex args?).
- [x] `expintx(ν,z) = exp(z) * expint(ν, z)`: #249
- [x] log of the incomplete gamma function log Γ(a,x), using `expintx` as commented [here](https://github.com/JuliaMath/SpecialFunctions.jl/pull/271#issuecomment-739011104): #283

(Even for real arguments, the `expint` implementation seems to be faster than what we have now for `gamma_inc`, although it might not be as accurate for the "lower" incomplete gamma function. I feel like we should just have a `gamma(n,x)` method that returns the upper incomplete gamma function, similar to Mathematica, in any case.)

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