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Plot of function (top) and function (bottom).

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.

Contents

DefinitionsEdit

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]

 

(note that 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[3]

 

The behaviour of E1 near the branch cut can be seen by the following relation:[4]

 

PropertiesEdit

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

Convergent seriesEdit

For real or complex arguments off the negative real axis,   can be expressed as[5]

 

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, e.g.:

 

for  .

Asymptotic (divergent) seriesEdit

 
Relative error of the asymptotic approximation for different number   of terms in the truncated sum

Unfortunately, the convergence of the series above is slow for arguments of larger modulus. For example, for x = 10 more than 40 terms are required to get an answer correct to three significant figures for  .[6] However, there is a divergent series approximation that can be obtained by integrating   by parts:[7]

 

which has error of order   and is valid for large values of  . 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).

Exponential and logarithmic behavior: bracketingEdit

 
Bracketing of   by elementary functions

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:[8]

 

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 EinEdit

Both   and   can be written more simply using the entire function  [9] defined as

 

(note that this is just the alternating series in the above definition of  ). Then we have

 
 

Relation with other functionsEdit

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(−z). In fact,

 

with the derivative evaluated at   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 incomplete gamma function:[10]

 

The generalized form is sometimes called the Misra function[11]  , defined as

 

Including a logarithm defines the generalized integro-exponential function[12]

 

The indefinite integral:

 

is similar in form to the ordinary generating function for  , the number of divisors of  :

 

DerivativesEdit

The derivatives of the generalised functions   can be calculated by means of the formula [13]

 

Note that the function   is easy to evaluate (making this recursion useful), since it is just  .[14]

Exponential integral of imaginary argumentEdit

 
  against  ; real part black, imaginary part red.

If   is imaginary, it has a nonnegative real part, so we can use the formula

 

to get a relation with the trigonometric integrals   and  :

 

The real and imaginary parts of   are plotted in the figure to the right with black and red curves.

ApproximationsEdit

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

  • The Swamee and Ohija approximation[15]
 
where
 
  • The Allen and Hastings approximation [15][16]
 
where
 
  • The continued fraction expansion [16]
 
  • The approximation of Barry et al. [17]
 
where:
 
with   being the Euler–Mascheroni constant.

ApplicationsEdit

  • Time-dependent heat transfer
  • Nonequilibrium groundwater flow in the Theis solution (called a well function)
  • Radiative transfer in stellar and planetary atmospheres
  • Radial diffusivity equation for transient or unsteady state flow with line sources and sinks
  • Solutions to the neutron transport equation in simplified 1-D geometries.[18]

See alsoEdit

NotesEdit

  1. ^ Abramowitz and Stegun, p. 228
  2. ^ Abramowitz and Stegun, p. 228, 5.1.1
  3. ^ Abramowitz and Stegun, p. 228, 5.1.4 with n = 1
  4. ^ Abramowitz and Stegun, p. 228, 5.1.7
  5. ^ Abramowitz and Stegun, p. 229, 5.1.11
  6. ^ Bleistein and Handelsman, p. 2
  7. ^ Bleistein and Handelsman, p. 3
  8. ^ Abramowitz and Stegun, p. 229, 5.1.20
  9. ^ Abramowitz and Stegun, p. 228, see footnote 3.
  10. ^ Abramowitz and Stegun, p. 230, 5.1.45
  11. ^ After Misra (1940), p. 178
  12. ^ Milgram (1985)
  13. ^ Abramowitz and Stegun, p. 230, 5.1.26
  14. ^ Abramowitz and Stegun, p. 229, 5.1.24
  15. ^ a b Giao, Pham Huy (2003-05-01). "Revisit of Well Function Approximation and An Easy Graphical Curve Matching Technique for Theis' Solution". Ground Water. 41 (3): 387–390. doi:10.1111/j.1745-6584.2003.tb02608.x. ISSN 1745-6584.
  16. ^ a b Tseng, Peng-Hsiang; Lee, Tien-Chang (1998-02-26). "Numerical evaluation of exponential integral: Theis well function approximation". Journal of Hydrology. 205 (1–2): 38–51. Bibcode:1998JHyd..205...38T. doi:10.1016/S0022-1694(97)00134-0.
  17. ^ Barry, D. A; Parlange, J. -Y; Li, L (2000-01-31). "Approximation for the exponential integral (Theis well function)". Journal of Hydrology. 227 (1–4): 287–291. Bibcode:2000JHyd..227..287B. doi:10.1016/S0022-1694(99)00184-5.
  18. ^ George I. Bell; Samuel Glasstone (1970). Nuclear Reactor Theory. Van Nostrand Reinhold Company.

ReferencesEdit

External linksEdit