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,
Plot of the exponential integral function Ei(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D
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.:
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 ( in red, in pink).
Asymptotics beyond all orders
Using integration by parts, we can obtain an explicit formula
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
(note that this is just the alternating series in the above definition of ). Then we have
Relation with other functions
is usually solved by the confluent hypergeometric functions and But when and that is,
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 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 NIST Digital Library of Mathematical Functions.
Including a logarithm defines the generalized integro-exponential function
The indefinite integral:
is similar in form to the ordinary generating function for , the number of divisors of :
The derivatives of the generalised functions can be calculated by means of the formula 
Note that the function is easy to evaluate (making this recursion useful), since it is just .
Exponential integral of imaginary argument
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.
There have been a number of approximations for the exponential integral function. These include:
- The Swamee and Ohija approximation where
- The Allen and Hastings approximation  where
- The continued fraction expansion 
- The approximation of Barry et al.  where:with being the Euler–Mascheroni constant.
- 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
- ^ Abramowitz and Stegun, p. 228
- ^ Abramowitz and Stegun, p. 228, 5.1.1
- ^ Abramowitz and Stegun, p. 228, 5.1.4 with n = 1
- ^ Abramowitz and Stegun, p. 228, 5.1.7
- ^ Abramowitz and Stegun, p. 229, 5.1.11
- ^ Bleistein and Handelsman, p. 2
- ^ Bleistein and Handelsman, p. 3
- ^ O’Malley, Robert E. (2014), O’Malley, Robert E. (ed.), “Asymptotic Approximations”, Historical Developments in Singular Perturbations, Cham: Springer International Publishing, pp. 27–51, doi:10.1007/978-3-319-11924-3_2, ISBN 978-3-319-11924-3, retrieved 2023-05-04
- ^ Abramowitz and Stegun, p. 229, 5.1.20
- ^ Abramowitz and Stegun, p. 228, see footnote 3.
- ^ Abramowitz and Stegun, p. 230, 5.1.45
- ^ After Misra (1940), p. 178
- ^ Milgram (1985)
- ^ Abramowitz and Stegun, p. 230, 5.1.26
- ^ Abramowitz and Stegun, p. 229, 5.1.24
- ^ 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. PMID 12772832. S2CID 31982931.
- ^ 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.
- ^ 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.
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