Derivatives of exponential integrals

[oxinabox]

Moving this from JuliaDiff/ChainRules.jl#292 since the code now lives here.

@stevengj said:

Just added to SpecialFunctions.jl: #236

The derivative with respect to x is a simple recurrence: https://en.wikipedia.org/wiki/Exponential_integral#Derivatives

[stevengj]

Closing based on #328. (Derivatives with respect to the order aren't going to happen for a long time, if ever…)

[giordano]

Derivatives with respect to the order aren't going to happen for a long time

Out of curiosity, what'd be the meaning of the derivative with respect to a discrete parameter? How one would go about that? By extending the domain from N to R?

[stevengj]

Out of curiosity, what'd be the meaning of the derivative with respect to a discrete parameter? How one would go about that? By extending the domain from N to R?

It's not a discrete parameter. Arbitrary complex order is well defined and supported:

julia> expint(1, 3)
0.013048381094197039

julia> expint(1.01, 3)
0.01301964756083072

julia> expint(1.01+0.4im, 3)
0.012931671071467582 - 0.0011410591248868325im

The extension isn't even very complicated — the definition can be straightforwardly evaluated for non-integer order

[giordano]

Oh, ok, I didn't check, but the name "order" made me think it was a discrete order. Thanks!