Plot of E₁ function (top) and Ei function (bottom).

Definitions

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

\mbox{Ei}(x)=\int_{-\infty}^x\frac{e^t}t\,dt.\,

The definition above can be used for positive values of x, but integral has to be understood in terms of the Cauchy principal value.

For complex values of the argument, the definition becomes ambiguous ^[1] (see http://www.math.sfu.ca/~cbm/aands/page_228.htm, formula 5.1.1); in order to avoid confusion, the following notation is used:

{\rm E}_1(z) = \int_z^\infty \frac{e^{-t}}{t} dt,\qquad|{\rm Arg}(z)|<\pi.

At positive values of real part of argument, this presentation can be converted to

{\rm E}_1(z) = \int_1^\infty \frac{e^{-tz}}{t}\, dt,\qquad \Re(z) \ge 0.

The function Ei is related with E₁ as follows:

\lim_{\delta\to0\pm}{\rm Ei}(-x+{\rm i}\delta) = - {\rm E}_1(x) \mp {\rm i} \pi,\qquad x>0,

-{\rm Ei}(x) = \frac{1}{2}\lim_{\delta\to0+}{\rm E}_1(-x+{\rm i}\delta) + \frac{1}{2}\lim_{\delta\to0-}{\rm E}_1(-x+{\rm i}\delta), \qquad x>0.

The extension of Ei to the complex plane may have cut at the negative values of argument. Then, area of analyticity of function Ei is complementary to that of E₁.

Properties

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

Convergent series

E₁ has logarithmic peculiarity at zero. The extraction allows to write the exponential integral in terms of convergent series:

\mbox{Ei}(x) = \gamma+\ln x+ \sum_{k=1}^{\infty} \frac{x^k}{k\; k!} \,, ~~~~~x>0

E_1(z) =-\gamma-\ln z+ \sum_{k=1}^{\infty} \frac{(-1)^{k+1} z^k}{k\; k!} \,,~~~~~~~~ {\rm Re}(z)>0

where ~\gamma\approx 0.5772156649015328606...~ is the Euler gamma constant. The series converges at any complex value of the argument, but definition of Ei requires that ~x\!>\!0~.

Asymptotic (divergent) series

Relative error of the asymptotic approximation for different number ~N~ of term in the truncated sum

E_1(z)=\frac{\exp(-z)}{z} \left( \sum_{n=0}^{N-1} \frac{n!}{(-z)^n} + {\mathcal{O}}\left( \frac{N!}{z^N} \right) \right)

The truncated sum can be used for evaluation at ~{\rm Re }(z)\!\gg\! 1~. The more terms are taken into account in the sum, the larger should be the real part of the argument in order to make the truncated sum useful for the evaluation.

The relative error of the approximation above is plotted on the figure. With truncated series, the function E_1(z,N)= \frac{\exp(-z)}{z} \sum_{n=0}^{N-1} \frac{n!}{(-z)^n} approximates E_1(z) at ~{\rm Re}(z) \gg 1. The relative error R(x)=\frac{ E_1(x,N)-E_1(x) }{E_1(x)} is plotted versus ~x~ for 3<x<9 for
~N=1~ (red),
~N=2~ (green),
~N=3~ (yellow),
~N=4~ (blue), ~N=5~ (pink). The larger is ~x~, the more terms in this expansion can be taken into account for approximation of the function (and the better is the resulting approximation). At ~x\!>\!40~, the approximation E_1(z,40) is preciser, that the direct evaluation of the initial integral with standard (double) variables in most of programming languages. The modulus of the last term taken into account characterizes the error of such approximation.

Exponential and logarithmic behavior: bracketing

Bracketing of ~ E_1(x)~ (black) with elementary functions (red and blue)

\frac{\exp(-x)}{2}\!~\ln\!\left(1+\frac{2}{x}\right) <E_1(x)< \exp(-x)\!~\ln\!\left(1+\frac{1}{x} \right) ~~~~~~~~x\!>\!0

The left-hand side of this inequality is shown in the Figure with red curve. The central part ~{\rm E}_1(x)~ is shown with the black curve. The right-hand side is shown with blue curve.

Relation with other functions

The exponential integral is closely related to the logarithmic integral function li(x),

li(x) = Ei (ln (x))    for all positive real x ≠ 1.

Also closely related is a function which integrates over a different range:

{\rm E}_1(x) = \int_1^\infty \frac{e^{-tx}}{t}\, dt = \int_x^\infty \frac{e^{-t}}{t} dt.

This function may be regarded as extending the exponential integral to the negative reals by

{\rm Ei}(-x) = - {\rm E}_1(x).\,

We can express both of them in terms of an entire function,

{\rm Ein}(x) = \int_0^x (1-e^{-t})\frac{dt}{t} = \sum_{k=1}^\infty \frac{(-1)^{k+1}x^k}{k\; k!}.

Using this function, we then may define, using the logarithm,

{\rm E}_1(z) \,=\, -\gamma-\ln z + {\rm Ein}(z),~~~~~~|{\rm Arg}(z)|<\pi~

and

{\rm Ei}(x) \,=\, \gamma+\ln x - {\rm Ein}(-x),~~~~~~x>0.

The exponential integral may also be generalized to

{\rm E}_n(x) = \int_1^\infty \frac{e^{-xt}}{t^n}\, dt,

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

{\rm E}_n(x) =x^{n-1}\Gamma(1-n,x).\,

The generalized form is sometimes called the Misra function \varphi_m(x), defined as

\varphi_m(x)={\rm E}_{-m}(x).\,

Derivatives

Functions ~{\rm E}_n~ are simply related with derivatives of ~{\rm E}_1~:

{{\rm E}_n} '(z) = -{\rm E}_{n-1}(z),~~~~~~~~(|{\rm Arg}(z)|<\pi,~~~ n>0)

However, ~n~ is supposed to be integer; the generalization for complex ~n~ is not yet reported in the literature, although the definition of ~{\rm E}_n~ through the integral could allow such a generalization.

Exponential integral of imaginary argument

{\rm E}_1( {\rm i}\!~ x) versus ~x~, real part(black) and imaginary part (red).

{\rm E}_1(z) = \int_1^\infty \frac {\exp(-zt)} {t} \,{\rm d} t, ~~~~~~({\rm Re}(z) \ge 0)

the relation of the exponential integral with integral sinus (Si) and integral cosinus (Ci) is seen:

{\rm E}_1( {\rm i}\!~ x)= -\frac{\pi}{2} +{\rm Si}(x)-{\rm i}\cdot {\rm Ci}(x),~~~~~~~~~(x>0)

Real and imaginary parts of ~{\rm E}_1(x)~ are plotted in Figure with black and red curves. The real part has logarithmic peculiarity at zero (As the Exponential integral of the real argument).

Integrals with exponential integral

Applications

• Time-dependent heat transfer
• Nonequilibrium groundwater 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

Notes

1. ↑

References

• Press, William H. et al. Numerical Recipes (FORTRAN). Cambridge University Press, New York: 1989.
• Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. (See Chapter 5)

• R. D. Misra, Proc. Cambridge Phil. Soc. 36, 173 (1940)
• S. Chandrasekhar, Radiative transfer, reprinted 1960, Dover

External links

• Formulas and identities for Ei

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