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Exponential integral

Exponential integral

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In mathematics, the exponential integral is a special function defined on the complex plane
Complex plane
In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...

 given the symbol Ei.

Definitions


For real, nonzero values of x, the exponential integral Ei(x) can be defined as


The function is given as a special function because is not an elementary function, a fact which can be proven using the Risch Algorithm
Risch algorithm
The Risch algorithm, named after Robert Henry Risch, is an algorithm for the calculus operation of indefinite integration . The algorithm transforms the problem of integration into a problem in algebra. It is based on the form of the function being integrated and on methods for integrating rational...

. 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
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:...

, due to the singularity in the integrand at zero. For complex values of the argument, the definition becomes ambiguous due to branch points at 0 and . In general, a branch cut is taken on the negative real axis and Ei can be defined by analytic continuation
Analytic continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which...

 elsewhere on the complex plane.

The following notation is used,


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:

Properties


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

Convergent series


Integrating the Taylor series
Taylor series
In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....

 for , and extracting the logarithmic singularity, we can derive the following series representation for for real :


For complex arguments off the negative real axis, this generalises to


where is the Euler–Mascheroni constant
Euler–Mascheroni constant
The Euler–Mascheroni constant is a mathematical constant recurring in analysis and number theory, usually denoted by the lowercase Greek letter ....

. The sum converges for all complex , and we take the usual value of the complex logarithm
Complex logarithm
In complex analysis, a complex logarithm function is an "inverse" of the complex exponential function, just as the natural logarithm ln x is the inverse of the real exponential function ex. Thus, a logarithm of z is a complex number w such that ew = z. The notation for such a w is log z...

 having a branch cut along the negative real axis.

Asymptotic (divergent) series


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. However, there is a divergent series approximation that can be obtained by integrating by parts:
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 ( in red, in pink). When , the approximation above with is correct to within 64 bit double precision
Double precision
In computing, double precision is a computer number format that occupies two adjacent storage locations in computer memory. A double-precision number, sometimes simply called a double, may be defined to be an integer, fixed point, or floating point .Modern computers with 32-bit storage locations...

.

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


Both and can be written more simply using the entire function
Entire function
In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic over the whole complex plane...

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

Relation with other functions


The exponential integral is closely related to the logarithmic integral function
Logarithmic integral function
In mathematics, the logarithmic integral function or integral logarithm li is a special function. It occurs in problems of physics and has number theoretic significance, occurring in the prime number theorem as an estimate of the number of prime numbers less than a given value.-Integral...

 li(x) by the formula
for positive real values of

The exponential integral may also be generalized to


which can be written as a special case of the incomplete gamma function
Incomplete gamma function
In mathematics, the gamma function is defined by a definite integral. The incomplete gamma function is defined as an integral function of the same integrand. There are two varieties of the incomplete gamma function: the upper incomplete gamma function is for the case that the lower limit of...

:


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


Including a logarithm defines the generalized integro-exponential function.

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

Applications

  • Time-dependent heat transfer
    Heat transfer
    Heat transfer is a discipline of thermal engineering that concerns the exchange of thermal energy from one physical system to another. Heat transfer is classified into various mechanisms, such as heat conduction, convection, thermal radiation, and phase-change transfer...

  • Nonequilibrium groundwater
    Groundwater
    Groundwater is water located beneath the ground surface in soil pore spaces and in the fractures of rock formations. A unit of rock or an unconsolidated deposit is called an aquifer when it can yield a usable quantity of water. The depth at which soil pore spaces or fractures and voids in rock...

     flow in the Theis solution (called a well function)
  • Radiative transfer in stellar atmospheres
  • Radial Diffusivity Equation for transient or unsteady state flow with line sources and sinks

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