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 − E 1 ( x ) = Ei ⁡ ( − x ) {displaystyle -E_{1}(x)=operatorname {Ei} (-x)} ). In general, a branch cut is taken on the negative real axis and E1 can be defined by analytic continuation elsewhere on the complex plane. For positive values of the real part of z {displaystyle z} , this can be written The behaviour of E1 near the branch cut can be seen by the following relation: