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Wikipedia / Exponential integral

In mathematics, the exponential integral 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, the exponential integral

The Risch algorithm shows that is not an elementary function. The definition above can be used for positive values of, 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 and . Instead of , the following notation is used,

For positive values of, we have .

In general, a branch cut is taken on the negative real axis and 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 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

can be expressed as 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.

between and . For , the result is inaccurate due to cancellation.

A faster converging series was found by Ramanujan:

Asymptotic (divergent) series

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 , there is a divergent series 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 pink).

Asymptotics beyond all orders

Using integration by parts, we can obtain an explicit formula For any fixed 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

can be bracketed by elementary functions as follows:

is shown in black and the right-hand side is shown in red.

Definition by Ein

can be written more simply using the entire function (note that this is just the alternating series in the above definition of is related to the exponential generating function of the harmonic numbers

Relation with other functions

is usually solved by the confluent hypergeometric functions and and for all . A second solution is then given by with the derivative evaluated at . Another connexion with the confluent hypergeometric functions is that is an exponential times the function

The exponential integral is closely related to the logarithmic integral function 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

which can be written as a special case of the upper incomplete gamma function:

The generalized form is sometimes called the Misra function ,

Including a logarithm defines the generalized integro-exponential function

Derivatives

can be calculated by means of the formula is easy to evaluate (making this recursion useful), since it is just .

Exponential integral of imaginary argument

to get a relation with the trigonometric integrals and are plotted in the figure to the right with black and red curves.

Approximations

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

Inverse function of the exponential integral

We can express the Inverse function of the exponential integral in power series form: is the Ramanujan–Soldner constant is polynomial sequence defined by the following recurrence relation

For

Applications

See also

Footnotes

Citations

References

  • [DLMF 8.19]