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.

Definitions

For real non-zero values of x, the exponential integral Ei 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:

Properties

Several properties of the exponential integral below, in certain cases, allow one to avoid its explicit evaluation through the definition above.

Convergent series

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:

Asymptotic (divergent) series

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.

Asymptotics beyond all orders

Using integration by parts, we can obtain an explicit formula For any fixed, the absolute value of the error term decreases, then increases. The minimum occurs at, at which point. This bound is said to be "asymptotics beyond all orders".

Exponential and logarithmic behavior: bracketing

From the two series suggested in previous subsections, it follows that behaves like a negative exponential for large values of the argument and like a logarithm for small values. For positive real values of the argument, can be bracketed by elementary functions as follows:
The left-hand side of this inequality is shown in the graph to the left in blue; the central part is shown in black and the right-hand side is shown in red.

Definition by Ein

Both and can be written more simply using the entire function defined as
. Then we have
The function is related to the exponential generating function of the harmonic numbers:

Relation with other functions

Kummer's equation
is usually solved by the confluent hypergeometric functions and But when and that is,
we have
for all z. A second solution is then given by E₁. In fact,
with the derivative evaluated at Another connexion with the confluent hypergeometric functions is that E₁ is an exponential times the function U:
The exponential integral is closely related to the logarithmic integral function li by the formula
for non-zero real values of.
The series expansion of the exponential integral immediately gives rise to an expression in terms of the generalized hypergeometric function :

Generalization

The exponential integral may also be generalized to
which can be written as a special case of the upper incomplete gamma function:
The generalized form is sometimes called the Misra function, defined as
Many properties of this generalized form can be found in the
Including a logarithm defines the generalized integro-exponential function

Derivatives

The derivatives of the generalised functions can be calculated by means of the formula
Note that the function is easy to evaluate, since it is just.