Hi Yep

The function

(exp x -1)/x = sum_0^inf x^n/(n+1)!

is an entire function so all derivatives and integrals are entire functions, too.

In closed form, the integral Ein(x) is represented as the sum of the integral of its summands Ei and Log.

Both functions Ei and Log have a logarithmic branch cut along the negative real axis, conventionally. So formally the singularities cancel. Nevertheless the fomula is generally valid for Re z > 0 only, because its parts don't allow any high school algeba along the branch cut.

For arguments with negative real part and small imaginary part one has to deal with the explicit Taylor series

sum_0^inf x^(n+1)/((n+1)(n+1)!)

which is rapidly convergent and can be approximated by PadeApproximants readily .

Regards Roland

Thanks but I do not get how your reply answered my question. First, z is not necessarily real; Second, Series produces a multi-valued expansion when it should not.