Exponential integral

Definitions

For real non-zero values of x, the exponential integral Ei(x) is defined as

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

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 \infty. Instead of Ei, the following notation is used,

:E_1(z) = \int_z^\infty \frac{e^{-t}}{t}, dt,\qquad|{\rm Arg}(z)|[[File: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.svg|alt=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|thumb|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 z, this can be written :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+} E_1(-x \pm i\delta) = -\operatorname{Ei}(x) \mp i\pi,\qquad x0.

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, E_1(z) can be expressed as

:E_1(z) = -\gamma - \ln z - \sum_{k=1}^{\infty} \frac{(-z)^k}{k; k!} \qquad (\left| \operatorname{Arg}(z) \right|

where \gamma 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 E_1(x) with floating point operations for real x between 0 and 2.5. For x 2.5, the result is inaccurate due to cancellation.

A faster converging series was found by Ramanujan:

:{\rm Ei} (x) = \gamma + \ln x + \exp{(x/2)} \sum_{n=1}^\infty \frac{ (-1)^{n-1} x^n} {n! , 2^{n-1}} \sum_{k=0}^{\lfloor (n-1)/2 \rfloor} \frac{1}{2k+1}

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 E_1(10). However, for positive values of x, there is a divergent series approximation that can be obtained by integrating x e^x E_1(x) by parts: : E_1(x)=\frac{\exp(-x)} x \left(\sum_{n=0}^{N-1} \frac{n!}{(-x)^n} +O(N!x^{-N}) \right) The relative error of the approximation above is plotted on the figure to the right for various values of N, the number of terms in the truncated sum (N=1 in red, N=5 in pink).

Asymptotics beyond all orders

Using integration by parts, we can obtain an explicit formula\operatorname{Ei}(z) = \frac{e^{z}} {z} \left (\sum {k=0}^{n} \frac{k!} {z^{k}} + e{n}(z)\right), \quad e_{n}(z) \equiv (n + 1)!\ ze^{-z}\int { -\infty }^{z} \frac{e^{t}} {t^{n+2}},dt For any fixed z, the absolute value of the error term |e_n(z)| decreases, then increases. The minimum occurs at n\sim |z|, at which point \vert e{n}(z)\vert \leq \sqrt{\frac{2\pi } {\vert z\vert }}e^{-\vert z\vert }. This bound is said to be "asymptotics beyond all orders".

Exponential and logarithmic behavior: bracketing

From the two series suggested in previous subsections, it follows that E_1 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, E_1 can be bracketed by elementary functions as follows: : \frac 1 2 e^{-x},\ln!\left( 1+\frac 2 x \right) \qquad x0

The left-hand side of this inequality is shown in the graph to the left in blue; the central part E_1(x) is shown in black and the right-hand side is shown in red.

Definition by Ein

Both \operatorname{Ei} and E_1 can be written more simply using the entire function \operatorname{Ein} defined as : \operatorname{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 E_1). Then we have : E_1(z) ,=, -\gamma-\ln z + {\rm Ein}(z) \qquad \left| \operatorname{Arg}(z) \right| :\operatorname{Ei}(x) ,=, \gamma+\ln{x} - \operatorname{Ein}(-x) \qquad x \neq 0 The function \operatorname{Ein} is related to the exponential generating function of the harmonic numbers: : \operatorname{Ein}(z) = e^{-z} , \sum_{n=1}^\infty \frac {z^n}{n!} H_n

Relation with other functions

Kummer's equation :z\frac{d^2w}{dz^2} + (b-z)\frac{dw}{dz} - aw = 0 is usually solved by the confluent hypergeometric functions M(a,b,z) and U(a,b,z). But when a=0 and b=1, that is, :z\frac{d^2w}{dz^2} + (1-z)\frac{dw}{dz} = 0 we have :M(0,1,z)=U(0,1,z)=1 for all z. A second solution is then given by E1(−z). In fact, :E_1(-z)=-\gamma-i\pi+\frac{\partial[U(a,1,z)-M(a,1,z)]}{\partial a},\qquad 0 with the derivative evaluated at a=0. Another connexion with the confluent hypergeometric functions is that E1 is an exponential times the function U(1,1,z): :E_1(z)=e^{-z}U(1,1,z)

The exponential integral is closely related to the logarithmic integral function li(x) by the formula :\operatorname{li}(e^x) = \operatorname{Ei}(x) for non-zero real values of x .

The series expansion of the exponential integral immediately gives rise to an expression in terms of the generalized hypergeometric function {}_2F_2: :\operatorname{Ei}(x) = x{}_2F_2(1,1;2,2;x)+\ln x+\gamma.

Generalization

The exponential integral may also be generalized to

:E_n(x) = \int_1^\infty \frac{e^{-xt}}{t^n}, dt,

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

: 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)=E_{-m}(x).

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

Derivatives

The derivatives of the generalised functions E_n can be calculated by means of the formula : E_n '(z) = - E_{n-1}(z) \qquad (n=1,2,3,\ldots) Note that the function E_0 is easy to evaluate (making this recursion useful), since it is just e^{-z}/z.

Exponential integral of imaginary argument

against x; real part black, imaginary part red.]]

If z is imaginary, it has a nonnegative real part, so we can use the formula : E_1(z) = \int_1^\infty \frac{e^{-tz}} t , dt to get a relation with the trigonometric integrals \operatorname{Si} and \operatorname{Ci}: : E_1(ix) = i\left[ -\tfrac{1}{2}\pi + \operatorname{Si}(x)\right] - \operatorname{Ci}(x) \qquad (x 0) The real and imaginary parts of \mathrm{E}_1(ix) 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 E_1(x) = \left (A^{-7.7}+B \right )^{-0.13}, where \begin{align} A &= \ln\left [\left (\frac{0.56146}{x}+0.65\right)(1+x)\right] \ B &= x^4e^{7.7x}(2+x)^{3.7} \end{align}
• The Allen and Hastings approximation E_1(x) = \begin{cases} - \ln x +\textbf{a}^T\textbf{x}_5,&x\leq1 \ \frac{e^{-x}} x \frac{\textbf{b}^T \textbf{x}_3}{\textbf{c}^T\textbf{x}_3},&x\geq1 \end{cases} where \begin{align} \textbf{a} & \triangleq [-0.57722, 0.99999, -0.24991, 0.05519, -0.00976, 0.00108]^T \ \textbf{b} & \triangleq[0.26777,8.63476, 18.05902, 8.57333]^T \ \textbf{c} & \triangleq[3.95850, 21.09965, 25.63296, 9.57332]^T \ \textbf{x}_k &\triangleq[x^0,x^1,\dots, x^k]^T \end{align}
• The continued fraction expansion E_1(x) = \cfrac{e^{-x}}{x+\cfrac{1}{1+\cfrac{1}{x+\cfrac{2}{1+\cfrac{2}{x+\cfrac{3}{\ddots}}}}}}.
• The approximation of Barry et al. E_1(x) = \frac{e^{-x}}{G+(1-G)e^{-\frac{x}{1-G}}}\ln\left[1+\frac G x -\frac{1-G}{(h+bx)^2}\right], where: \begin{align} h &= \frac{1}{1+x\sqrt{x}}+\frac{h_{\infty}q}{1+q} \ q &=\frac{20}{47}x^{\sqrt{\frac{31}{26}}} \ h_{\infty} &= \frac{(1-G)(G^2-6G+12)}{3G(2-G)^2b} \ b &=\sqrt{\frac{2(1-G)}{G(2-G)}} \ G &= e^{-\gamma} \end{align} with \gamma being the Euler–Mascheroni constant.

Inverse function of the Exponential Integral

We can express the Inverse function of the exponential integral in power series form:

: \forall |x|

where \mu is the Ramanujan–Soldner constant and (P_n) is polynomial sequence defined by the following recurrence relation:

: P_0(x) = x,\ P_{n+1}(x) = x(P_n'(x) - nP_n(x)).

For n 0, \deg P_n = n and we have the formula :

: P_n(x) = \left.\left(\frac{\mathrm d}{\mathrm dt}\right)^{n-1} \left(\frac{te^x}{\mathrm{Ei}(t+x)-\mathrm{Ei}(x)}\right)^n\right|_{t=0}.

Applications

• 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
• Solutions to the Trachenko-Zaccone nonlinear differential equation for the stretched exponential function in the relaxation of amorphous solids and glass transition

Notes

References

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References

1. Abramowitz and Stegun, p. 228
2. Abramowitz and Stegun, p. 228, 5.1.1
3. Abramowitz and Stegun, p. 228, 5.1.4 with ''n'' = 1
4. Abramowitz and Stegun, p. 228, 5.1.7
5. Abramowitz and Stegun, p. 229, 5.1.11
6. Andrews and Berndt, p. 130, 24.16
7. Bleistein and Handelsman, p. 2
8. Bleistein and Handelsman, p. 3
9. O’Malley, Robert E.. (2014). "Asymptotic Approximations". Springer International Publishing.
10. Abramowitz and Stegun, p. 229, 5.1.20
11. Abramowitz and Stegun, p. 228, see footnote 3.
12. Abramowitz and Stegun, p. 230, 5.1.45
13. After Misra (1940), p. 178
14. Milgram (1985)
15. Abramowitz and Stegun, p. 230, 5.1.26
16. Abramowitz and Stegun, p. 229, 5.1.24
17. Giao, Pham Huy. (2003-05-01). "Revisit of Well Function Approximation and An Easy Graphical Curve Matching Technique for Theis' Solution". Ground Water.
18. (1998-02-26). "Numerical evaluation of exponential integral: Theis well function approximation". Journal of Hydrology.
19. (2000-01-31). "Approximation for the exponential integral (Theis well function)". Journal of Hydrology.
20. "Inverse function of the Exponential Integral {{math".
21. George I. Bell. (1970). "Nuclear Reactor Theory". Van Nostrand Reinhold Company.
22. (2021-06-14). "Slow stretched-exponential and fast compressed-exponential relaxation from local event dynamics". Journal of Physics: Condensed Matter.
23. (2024-02-23). "Unifying Physical Framework for Stretched-Exponential, Compressed-Exponential, and Logarithmic Relaxation Phenomena in Glassy Polymers". Macromolecules.