In mathematics, the exponential integral Ei is a
special function
Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications.
The term is defined by ...
on the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
.
It is defined as one particular
definite integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with di ...
of the ratio between an
exponential function
The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, a ...
and its
argument
An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialectic ...
.
Definitions
For real non-zero values of ''x'', the exponential integral Ei(''x'') is defined as
:
The
Risch algorithm
In symbolic computation, the Risch algorithm is a method of indefinite integration used in some computer algebra systems to find antiderivatives. It is named after the American mathematician Robert Henry Risch, a specialist in computer algebra ...
shows that Ei is not an
elementary function
In mathematics, an elementary function is a function of a single variable (typically real or complex) that is defined as taking sums, products, roots and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, and exponen ...
. 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
In mathematics, the Cauchy principal value, named after Augustin Louis Cauchy, is a method for assigning values to certain improper integrals which would otherwise be undefined.
Formulation
Depending on the type of singularity in the integrand , ...
due to the singularity of the integrand at zero.
For complex values of the argument, the definition becomes ambiguous due to
branch points
In the mathematical field of complex analysis, a branch point of a multi-valued function (usually referred to as a "multifunction" in the context of complex analysis) is a point such that if the function is n-valued (has n values) at that point, ...
at 0 and Instead of Ei, the following notation is used,
:
For positive values of ''x'', we have
In general, a
branch cut
In the mathematical field of complex analysis, a branch point of a multi-valued function (usually referred to as a "multifunction" in the context of complex analysis) is a point such that if the function is n-valued (has n values) at that point, a ...
is taken on the negative real axis and ''E''
1 can be defined by
analytic continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new ...
elsewhere on the complex plane.
For positive values of the real part of
, this can be written
:
The behaviour of ''E''
1 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
Euler's constant (sometimes also called the Euler–Mascheroni constant) is a mathematical constant usually denoted by the lowercase Greek letter gamma ().
It is defined as the limiting difference between the harmonic series and the natural l ...
. The sum converges for all complex
, and we take the usual value of the
complex logarithm
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
having a
branch cut
In the mathematical field of complex analysis, a branch point of a multi-valued function (usually referred to as a "multifunction" in the context of complex analysis) is a point such that if the function is n-valued (has n values) at that point, a ...
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. :
:
for
.
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).
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
In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any fin ...
defined as
:
(note that this is just the alternating series in the above definition of
). Then we have
:
:
Relation with other functions
Kummer's equation
:
is usually solved by the
confluent hypergeometric functions
In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregular ...
and
But when
and
that is,
:
we have
:
for all ''z''. A second solution is then given by E
1(−''z''). In fact,
:
with the derivative evaluated at
Another connexion with the confluent hypergeometric functions is that ''E
1'' is an exponential times the function ''U''(1,1,''z''):
:
The exponential integral is closely related to the
logarithmic integral function
In mathematics, the logarithmic integral function or integral logarithm li(''x'') is a special function. It is relevant in problems of physics and has number theoretic significance. In particular, according to the prime number theorem, it is a ...
li(''x'') by the formula
:
for non-zero real values of
.
Generalization
The exponential integral may also be generalized to
:
which can be written as a special case of the upper
incomplete gamma function
In mathematics, the upper and lower incomplete gamma functions are types of special functions which arise as solutions to various mathematical problems such as certain integrals.
Their respective names stem from their integral definitions, which ...
:
:
The generalized form is sometimes called the Misra function
, defined as
:
Many properties of this generalized form can be found in th
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
In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary seri ...
for
, the number of
divisors
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
of
:
:
Derivatives
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 integral
In mathematics, trigonometric integrals are a indexed family, family of integrals involving trigonometric functions.
Sine integral
The different sine integral definitions are
\operatorname(x) = \int_0^x\frac\,dt
\operatorname(x) = -\int ...
s
and
:
:
The real and imaginary parts of
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:
* 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
Euler's constant (sometimes also called the Euler–Mascheroni constant) is a mathematical constant usually denoted by the lowercase Greek letter gamma ().
It is defined as the limiting difference between the harmonic series and the natural l ...
.
Applications
* Time-dependent
heat transfer
Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy (heat) between physical systems. Heat transfer is classified into various mechanisms, such as thermal conduction, ...
* Nonequilibrium
groundwater
Groundwater is the water present beneath Earth's surface in rock and soil pore spaces and in the fractures of rock formations. About 30 percent of all readily available freshwater in the world is groundwater. A unit of rock or an unconsolidate ...
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
Neutron transport (also known as neutronics) is the study of the motions and interactions of neutrons with materials. Nuclear scientists and engineers often need to know where neutrons are in an apparatus, what direction they are going, and how qu ...
equation in simplified 1-D geometries
See also
*
Goodwin–Staton integral In mathematics the Goodwin–Staton integral is defined as : Frank William John Olver (ed.), N. M. Temme (Chapter contr.), NIST Handbook of Mathematical Functions, Chapter 7, p160,Cambridge University Press 2010
: G(z)=\int_0^\infty \frac \, dt
...
*
Bickley–Naylor functions
Notes
References
*
Chapter 5
*
*
*
*
*
*
*
*
*
*
*
*
*
External links
*
NIST documentation on the Generalized Exponential Integral*
*
*
Exponential, Logarithmic, Sine, and Cosine Integralsin
DLMF.
{{DEFAULTSORT:Exponential Integral
Exponentials
Special functions
Special hypergeometric functions
Integrals