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Concept
Exponential integral
Summary
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 . Instead of Ei, the following notation is used,
For positive values of x, we have .
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 , this can be written
The behaviour of E1 near the branch cut can be seen by the following relation:
Several properties of the exponential integral below, in certain cases, allow one to avoid its explicit evaluation through the definition above.
For real or complex arguments off the negative real axis, can be expressed as
where is the Euler–Mascheroni constant. The sum converges for all complex , 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 with floating point operations for real between 0 and 2.5. For , the result is inaccurate due to cancellation.
A faster converging series was found by Ramanujan:
These alternating series can also be used to give good asymptotic bounds for small x, e.g. :
for .
Unfortunately, the convergence of the series above is slow for arguments of larger modulus. For example, more than 40 terms are required to get an answer correct to three significant figures for . However, for positive values of x, there is a divergent series approximation that can be obtained by integrating by parts:
The relative error of the approximation above is plotted on the figure to the right for various values of , the number of terms in the truncated sum ( in red, in pink).
Official source
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