Definitions
For real nonzero values of x, the exponential integral Ei(x) is defined as
The Risch Algorithm shows that Ei is not an elementary function. The definition above can be used for positive values of x, but the integral has to be understood in terms of the Cauchy principal value due to the singularity of the integrand at zero. For complex values of the argument, the definition becomes ambiguous due to branch points at 0 and . In general, a branch cut is taken on the negative real axis and Ei can be defined by analytic continuation elsewhere on the complex plane.
The following notation is used,
For positive values of the real part of, this can be written
The behaviour of E1 near the branch cut can be seen by the following relation:
Read more about this topic: Exponential Integral
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