Exponential integral

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

In mathematics, the exponential integral is a special function defined on the complex plane given the symbol Ei.

Definitions

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

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

The function is given as a special function because $\textstyle\int \frac{e^t}{t}\,dt$ is not an elementary function, a fact which can be proven using the Risch Algorithm. 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 in the integrand at zero. For complex values of the argument, the definition becomes ambiguous due to branch points at 0 and $\infty$. In general, a branch cut is taken on the negative real axis and Ei can be defined by analytic continuation elsewhere on the complex plane.

The following notation is used,

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

For positive values of the real part of z, this can be written

$\mathrm{E}_1(z) = \int_1^\infty \frac{e^{-tz}}{t}\, dt = \int_0^1 \frac{e^{-z/u}}{u}\, du ,\qquad \Re(z) \ge 0.$

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

$\lim_{\delta\to0\pm}\mathrm{E_1}(-x+i\delta) = -\mathrm{Ei}(x) \mp i\pi,\qquad x>0,$

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 for e ^{− t} / t, and extracting the logarithmic singularity, we can derive the following series representation for E₁(x) for real x:

$\mathrm{Ei}(x) = \gamma+\ln |x| + \sum_{k=1}^{\infty} \frac{x^k}{k\; k!} \qquad x \neq 0$

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

$\mathrm{E_1}(z) =-\gamma-\ln z+\sum_{k=1}^{\infty}\frac{(-1)^{k+1} z^k}{k\; k!} \qquad (|\mathrm{Arg}(z)| < \pi)$

where γ is the Euler–Mascheroni constant. The sum converges for all complex z, 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 E₁(x) with floating point operations for real x between 0 and 2.5. For x > 2.5, the result is inaccurate due to cancellation.

Asymptotic (divergent) series

Relative error of the asymptotic approximation for different number $~N~$ of term 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. However, there is a divergent series approximation that can be obtained by integrating ze^zE₁(z) by parts:

$\mathrm{E_1}(z)=\frac{\exp(-z)}{z}\sum_{n=0}^{N-1} \frac{n!}{(-z)^n}$

which has error of order O(N!z ^{− N}) and is valid for large values of Re(z). The relative error of the approximation above is plotted on the figure to the right for various values of N (N = 1 in red, N = 5 in pink). When x > 40, the approximation above with N = 40 is correct to within 64 bit double precision.

Computing medium values

To compute values for x between 2.5 and 40 in double precision, it is possible to use the following algorithm :

    Function MediumE1(ByVal x As Double) As Double
        If x < 2.5 Or x > 40 Then Return 0
        Dim result As Double = 1
        For n As Integer = 40 To 1 Step -1   '40 can be changed here to adjust precision
            result = 2 - 1 / result + x / n
        Next
        Return (1 - 1 / result) / x * Math.Exp(-x)
    End Function

Exponential and logarithmic behavior: bracketing

Bracketing of E₁ by elementary functions

From the two series suggested in previous subsections, it follows that E₁ 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, E₁ can be bracketed by elementary functions as follows:

$\frac{1}{2}e^{-x}\,\ln\!\left( 1+\frac{2}{x} \right) < \mathrm{E_1}(x) < e^{-x}\,\ln\!\left( 1+\frac{1}{x} \right) \qquad x>0$

The left-hand side of this inequality is shown in the graph to the left in blue; the central part E₁(x) is shown in black and the right-hand side is shown in red.

Definition by Ein

Both Ei and E₁ can be written more simply using the entire function Ein defined as

$\mathrm{Ein}(z) = \int_0^z (1-e^{-t})\frac{dt}{t} = \sum_{k=1}^\infty \frac{(-1)^{k+1}z^k}{k\; k!}$

(note that this is just the alternating series in the above definition of E₁). Then we have

$\mathrm{E_1}(z) \,=\, -\gamma-\ln z + {\rm Ein}(z) \qquad |\mathrm{Arg}(z)| < \pi$

$\mathrm{Ei}(x) \,=\, \gamma+\ln x - \mathrm{Ein}(-x) \qquad x>0$

Relation with other functions

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

$\mathrm{li}(x) = \mathrm{Ei}(\ln x)\,$

for positive real values of x

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 φ_m(x), defined as

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

Including a logarithm defines the generalized integro-exponential function

$E_s^j(z)= \frac{1}{\Gamma(j+1)}\int_1^\infty (\log t)^j \frac{e^{-zt}}{t^s}\,dt$.

Derivatives

The derivatives of the generalised functions E_n can be calculated by means of the formula

$\mathrm{E_n}'(z) = -\mathrm{E_{n-1}}(z) \qquad (n=1,2,3,\ldots)$

Note that the function E₀ is easy to evaluate (making this recursion useful), since it is just e ^{− z} / z.

Exponential integral of imaginary argument

E₁(ix) against x; real part black, imaginary part red.

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

$\mathrm{E_1}(z) = \int_1^\infty \frac{e^{-tz}}{t} dt$

to get a relation with the trigonometric integrals Si and Ci:

$\mathrm{E_1}(ix) = i\left(-\tfrac{1}{2}\pi + \mathrm{Si}(x)\right) - \mathrm{Ci}(x) \qquad (x>0)$

The real and imaginary parts of E₁(x) are plotted in the figure to the right with black and red curves.

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
• Solutions to the neutron transport equation in simplified 1-D geometries.

References

• Abramovitz, Milton; Irene Stegun (1964). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Abramowitz and Stegun. New York: Dover. ISBN 0-486-61272-4. http://www.math.sfu.ca/~cbm/aands. 
• Bender, Carl M.; Steven A. Orszag (1978). Advanced mathematical methods for scientists and engineers. McGraw-Hill. ISBN 0-07-004452-X. 
• Bleistein, Norman; Richard A. Handelsman (1986). Asymptotic Expansions of Integrals. Dover. ISBN 0486650820. 
• Busbridge, Ida W. (1950). "On the integro-exponential function and the evaluation of some integrals involving it". Quart. J. Math. (Oxford) 1 (1): 176-184. Bibcode 1950QJMat...1..176B. doi:10.1093/qmath/1.1.176. 
• Stankiewicz, A. (1968). "Tables of the integro-exponetial functions". Acta Astronomica 18: 289. Bibcode 1968AcA....18..289S. 
• Sharma, R. R.; Zohuri, Bahman (1977). "A general method for an accurate evaluation of exponential integrals E₁(x), x>0". J. Comput. Phys. 25 (2): 199—204. Bibcode 1977JCoPh..25..199S. doi:10.1016/0021-9991(77)90022-5. 
• Kölbig, K. S. (1983). "On the integral exp(-μt)t^ν-1log^mt dt". Math. Comput 41 (163): 171—182. doi:10.1090/S0025-5718-1983-0701632-1. 
• Milgram, M. S. (1985). "The generalized integro-exponential function". Mathematics of Computation 44 (170): 443–458. doi:10.1090/S0025-5718-1985-0777276-4. JSTOR 2007964. MR0777276. 
• Misra, Rama Dhar; Born, M. (1940). "On the Stability of Crystal Lattices. II". Mathematical Proceedings of the Cambridge Philosophical Society 36 (2): 173. doi:10.1017/S030500410001714X. 
• Chiccoli, C.; Lorenzutta, S.; Maino, G. (1988). "On the evaluation of generalized exponential integrals E_ν(x)". J. Comput. Phys. 78: 278—287. Bibcode 1988JCoPh..78..278C. doi:10.1016/0021-9991(88)90050-2. 
• Chiccoli, C.; Lorenzutta, S.; Maino, G. (1990). "Recent results for generalized exponential integrals". Computer Math. Applic. 19 (5): 21—29. doi:10.1016/0898-1221(90)90098-5. 
• MacLeod, Allan J. (2002). "The efficient computation of some generalised exponential integrals". J. Comput. Appl. Math. 148 (2): 363—374. Bibcode 2002JCoAm.138..363M. doi:10.1016/S0377-0427(02)00556-3. 
• Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), "Section 6.3. Exponential Integrals", Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, ISBN 978-0-521-88068-8, http://apps.nrbook.com/empanel/index.html#pg=266 

• Temme, N. M. (2010), "Exponential, Logarithmic, Sine, and Cosine Integrals", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F. et al., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR2723248, http://dlmf.nist.gov/6 

External links

• NIST documentation on the Generalized Exponential Integral
• Weisstein, Eric W., "Exponential Integral" from MathWorld.
• Weisstein, Eric W., "En-Function" from MathWorld.
• Formulas and identities for Ei