Editing Exponential integral (section)

==Definitions==
For real non-zero values of&nbsp;''x'', the exponential integral&nbsp;Ei(''x'') is defined as

:<math> \operatorname{Ei}(x) = -\int_{-x}^\infty \frac{e^{-t}}t\,dt = \int_{-\infty}^x \frac{e^t}t\,dt.</math>

The [[Risch algorithm]] shows that Ei is not an [[elementary function]].  The definition above can be used for positive values of&nbsp;''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 {{nowrap|<math>\infty</math>.}}<ref>Abramowitz and Stegun, p.&nbsp;228</ref> Instead of Ei, the following notation is used,<ref>Abramowitz and Stegun, p.&nbsp;228, 5.1.1</ref>

:<math>E_1(z) = \int_z^\infty \frac{e^{-t}}{t}\,  dt,\qquad|{\rm Arg}(z)|<\pi</math>[[File:Plot of the exponential integral function Ei(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D.svg|alt=Plot of the exponential integral function Ei(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D|thumb|Plot of the exponential integral function Ei(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D]]

For positive values of&nbsp;''x'', we have {{nowrap|<math>-E_1(x) = \operatorname{Ei}(-x)</math>.}}

In general, a [[branch cut]] is taken on the negative real axis and ''E''<sub>1</sub> can be defined by [[analytic continuation]] elsewhere on the complex plane.

For positive values of the real part of <math>z</math>, this can be written<ref>Abramowitz and Stegun, p.&nbsp;228, 5.1.4 with ''n''&nbsp;=&nbsp;1</ref>
:<math>E_1(z) = \int_1^\infty \frac{e^{-tz}}{t}\, dt = \int_0^1 \frac{e^{-z/u}}{u}\, du ,\qquad \Re(z) \ge 0.</math>

The behaviour of ''E''<sub>1</sub> near the branch cut can be seen by the following relation:<ref>Abramowitz and Stegun, p.&nbsp;228, 5.1.7</ref>

:<math>\lim_{\delta\to0+} E_1(-x \pm i\delta) = -\operatorname{Ei}(x) \mp i\pi,\qquad x>0.</math>