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.

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

[1] Instead of Ei, the following notation is used,[2] 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.

, this can be written[3] The behaviour of E1 near the branch cut can be seen by the following relation:[4] Several properties of the exponential integral below, in certain cases, allow one to avoid its explicit evaluation through the definition above.

, and we take the usual value of the complex logarithm having a branch cut along the negative real axis.

, the result is inaccurate due to cancellation.

A faster converging series was found by Ramanujan:[6] 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

[7] However, for positive values of x, there is a divergent series approximation that can be obtained by integrating

by parts:[8] 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 (

Using integration by parts, we can obtain an explicit formula[9]

For positive real values of the argument,

can be bracketed by elementary functions as follows:[10] The left-hand side of this inequality is shown in the graph to the left in blue; the central part

can be written more simply using the entire function

[11] defined as (note that this is just the alternating series in the above definition of

is related to the exponential generating function of the harmonic numbers: Kummer's equation is usually solved by the confluent hypergeometric functions

Another connexion with the confluent hypergeometric functions is that E1 is an exponential times the function U(1,1,z): The exponential integral is closely related to the logarithmic integral function li(x) by the formula for non-zero real values of

The exponential integral may also be generalized to which can be written as a special case of the upper incomplete gamma function:[12] The generalized form is sometimes called the Misra function[13]

, defined as Many properties of this generalized form can be found in the NIST Digital Library of Mathematical Functions.

can be calculated by means of the formula [15] Note that the function

is imaginary, it has a nonnegative real part, so we can use the formula to get a relation with the trigonometric integrals

are plotted in the figure to the right with black and red curves.

There have been a number of approximations for the exponential integral function.

These include: We can express the Inverse function of the exponential integral in power series form:[20] where

is polynomial sequence defined by the following recurrence relation: For