Editing Exponential integral (section)

===Convergent series===
[[Image:Exponential integral.svg|300px|right|thumb| Plot of <math>E_1</math> function (top) and <math>\operatorname{Ei}</math> function (bottom).]]

For real or complex arguments off the negative real axis, <math>E_1(z)</math> can be expressed as<ref>Abramowitz and Stegun, p.&nbsp;229, 5.1.11</ref>

:<math>E_1(z) = -\gamma - \ln z - \sum_{k=1}^{\infty} \frac{(-z)^k}{k\; k!} \qquad (\left| \operatorname{Arg}(z) \right| < \pi)</math>

where <math>\gamma</math> is the [[Euler–Mascheroni constant]]. The sum converges for all complex <math>z</math>, and we take the usual value of the [[complex logarithm]] having a [[branch cut]] along the negative real axis.

This formula can be used to compute <math>E_1(x)</math> with floating point operations for real <math>x</math> between 0 and 2.5. For <math>x > 2.5</math>, the result is inaccurate due to [[Catastrophic cancellation|cancellation]].

A faster converging series was found by [[Ramanujan]]:<ref>Andrews and Berndt, p.&nbsp;130, 24.16</ref>

:<math>{\rm Ei} (x) = \gamma + \ln x + \exp{(x/2)} \sum_{n=1}^\infty \frac{ (-1)^{n-1} x^n} {n! \, 2^{n-1}} \sum_{k=0}^{\lfloor (n-1)/2 \rfloor} \frac{1}{2k+1}</math>