Exponential integral

In mathematics, the exponential integral Ei is a special function on the complex plane. It is defined as one particular definite integral of the ratio between an exponential function and its argument.

For real non zero 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 ${\displaystyle \infty }$. Instead of Ei, the following notation is used,

(note that for positive values of  x, we have ${\displaystyle E_{1}(x)=-\operatorname {Ei} (-x)}$).

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