eint1

Function: eint1
Section: transcendental
C-Name: veceint1
Prototype: GDGp
Help: eint1(x,{n}): exponential integral E1(x). If n is present and x > 0,
 computes the vector of the first n values of the exponential integral E1(n x).
Doc: exponential integral $\int_x^\infty \dfrac{e^{-t}}{t}\,dt =
 \kbd{incgam}(0, x)$, where the latter expression extends the function
 definition from real $x > 0$ to all complex $x \neq 0$.

 If $n$ is present, we must have $x > 0$; the function returns the
 $n$-dimensional vector $[\kbd{eint1}(x),\dots,\kbd{eint1}(nx)]$. Contrary to
 other transcendental functions, and to the default case ($n$ omitted), the
 values are correct up to a bounded \emph{absolute}, rather than relative,
 error $10^{-n}$, where $n$ is \kbd{precision}$(x)$ if $x$ is a \typ{REAL}
 and defaults to \kbd{realprecision} otherwise. (In the most important
 application, to the computation of $L$-functions via approximate functional
 equations, those values appear as weights in long sums and small individual
 relative errors are less useful than controlling the absolute error.) This is
 faster than repeatedly calling \kbd{eint1($i$ * x)}, but less precise.

Variant: Also available is \fun{GEN}{eint1}{GEN x, long prec}.