Exponential integral

From HandWiki
Short description: Special function defined by an integral
Plot of the exponential integral function E n(z) with n=2 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D
Plot of the exponential integral function E n(z) with n=2 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D

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

It is defined as one particular definite integral of the ratio between an exponential function and its argument.

Definitions

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

[math]\displaystyle{ \operatorname{Ei}(x) = -\int_{-x}^\infty \frac{e^{-t}}t\,dt = \int_{-\infty}^x \frac{e^t}t\,dt. }[/math]

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 [math]\displaystyle{ \infty }[/math].[1] Instead of Ei, the following notation is used,[2]

[math]\displaystyle{ E_1(z) = \int_z^\infty \frac{e^{-t}}{t}\, dt,\qquad|{\rm Arg}(z)|\lt \pi }[/math]
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
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 [math]\displaystyle{ -E_1(x) = \operatorname{Ei}(-x) }[/math].

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 [math]\displaystyle{ z }[/math], this can be written[3]

[math]\displaystyle{ 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. }[/math]

The behaviour of E1 near the branch cut can be seen by the following relation:[4]

[math]\displaystyle{ \lim_{\delta\to0+} E_1(-x \pm i\delta) = -\operatorname{Ei}(x) \mp i\pi,\qquad x\gt 0. }[/math]

Properties

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

Convergent series

Plot of [math]\displaystyle{ E_1 }[/math] function (top) and [math]\displaystyle{ \operatorname{Ei} }[/math] function (bottom).

For real or complex arguments off the negative real axis, [math]\displaystyle{ E_1(z) }[/math] can be expressed as[5]

[math]\displaystyle{ E_1(z) = -\gamma - \ln z - \sum_{k=1}^{\infty} \frac{(-z)^k}{k\; k!} \qquad (\left| \operatorname{Arg}(z) \right| \lt \pi) }[/math]

where [math]\displaystyle{ \gamma }[/math] is the Euler–Mascheroni constant. The sum converges for all complex [math]\displaystyle{ z }[/math], 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 [math]\displaystyle{ E_1(x) }[/math] with floating point operations for real [math]\displaystyle{ x }[/math] between 0 and 2.5. For [math]\displaystyle{ x \gt 2.5 }[/math], the result is inaccurate due to cancellation.

A faster converging series was found by Ramanujan:

[math]\displaystyle{ {\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} }[/math]

These alternating series can also be used to give good asymptotic bounds for small x, e.g.[citation needed]:

[math]\displaystyle{ 1-\frac{3 x}{4}\le{\rm Ei} (x) - \gamma - \ln x \le 1-\frac{3 x}{4}+\frac{11 x^2}{36} }[/math]

for [math]\displaystyle{ x\ge 0 }[/math].

Asymptotic (divergent) series

Relative error of the asymptotic approximation for different number [math]\displaystyle{ ~N~ }[/math] of terms in the truncated sum

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 [math]\displaystyle{ E_1(10) }[/math].[6] However, for positive values of x, there is a divergent series approximation that can be obtained by integrating [math]\displaystyle{ x e^x E_1(x) }[/math] by parts:[7]

[math]\displaystyle{ E_1(x)=\frac{\exp(-x)} x \left(\sum_{n=0}^{N-1} \frac{n!}{(-x)^n} +O(N!x^{-N}) \right) }[/math]

The relative error of the approximation above is plotted on the figure to the right for various values of [math]\displaystyle{ N }[/math], the number of terms in the truncated sum ([math]\displaystyle{ N=1 }[/math] in red, [math]\displaystyle{ N=5 }[/math] in pink).

Asymptotics beyond all orders

Using integration by parts, we can obtain an explicit formula[8][math]\displaystyle{ \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 }[/math]For any fixed [math]\displaystyle{ z }[/math], the absolute value of the error term [math]\displaystyle{ |e_n(z)| }[/math] decreases, then increases. The minimum occurs at [math]\displaystyle{ n\sim |z| }[/math], at which point [math]\displaystyle{ \vert e_{n}(z)\vert \leq \sqrt{\frac{2\pi } {\vert z\vert }}e^{-\vert z\vert } }[/math]. This bound is said to be "asymptotics beyond all orders".

Exponential and logarithmic behavior: bracketing

Bracketing of [math]\displaystyle{ E_1 }[/math] by elementary functions

From the two series suggested in previous subsections, it follows that [math]\displaystyle{ E_1 }[/math] 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, [math]\displaystyle{ E_1 }[/math] can be bracketed by elementary functions as follows:[9]

[math]\displaystyle{ \frac 1 2 e^{-x}\,\ln\!\left( 1+\frac 2 x \right) \lt E_1(x) \lt e^{-x}\,\ln\!\left( 1+\frac 1 x \right) \qquad x\gt 0 }[/math]

The left-hand side of this inequality is shown in the graph to the left in blue; the central part [math]\displaystyle{ E_1(x) }[/math] is shown in black and the right-hand side is shown in red.

Definition by Ein

Both [math]\displaystyle{ \operatorname{Ei} }[/math] and [math]\displaystyle{ E_1 }[/math] can be written more simply using the entire function [math]\displaystyle{ \operatorname{Ein} }[/math][10] defined as

[math]\displaystyle{ \operatorname{Ein}(z) = \int_0^z (1-e^{-t})\frac{dt}{t} = \sum_{k=1}^\infty \frac{(-1)^{k+1}z^k}{k\; k!} }[/math]

(note that this is just the alternating series in the above definition of [math]\displaystyle{ \mathrm{E}_1 }[/math]). Then we have

[math]\displaystyle{ E_1(z) \,=\, -\gamma-\ln z + {\rm Ein}(z) \qquad \left| \operatorname{Arg}(z) \right| \lt \pi }[/math]
[math]\displaystyle{ \operatorname{Ei}(x) \,=\, \gamma+\ln{x} - \operatorname{Ein}(-x) \qquad x \neq 0 }[/math]

Relation with other functions

Kummer's equation

[math]\displaystyle{ z\frac{d^2w}{dz^2} + (b-z)\frac{dw}{dz} - aw = 0 }[/math]

is usually solved by the confluent hypergeometric functions [math]\displaystyle{ M(a,b,z) }[/math] and [math]\displaystyle{ U(a,b,z). }[/math] But when [math]\displaystyle{ a=0 }[/math] and [math]\displaystyle{ b=1, }[/math] that is,

[math]\displaystyle{ z\frac{d^2w}{dz^2} + (1-z)\frac{dw}{dz} = 0 }[/math]

we have

[math]\displaystyle{ M(0,1,z)=U(0,1,z)=1 }[/math]

for all z. A second solution is then given by E1(−z). In fact,

[math]\displaystyle{ E_1(-z)=-\gamma-i\pi+\frac{\partial[U(a,1,z)-M(a,1,z)]}{\partial a},\qquad 0\lt {\rm Arg}(z)\lt 2\pi }[/math]

with the derivative evaluated at [math]\displaystyle{ a=0. }[/math] Another connexion with the confluent hypergeometric functions is that E1 is an exponential times the function U(1,1,z):

[math]\displaystyle{ E_1(z)=e^{-z}U(1,1,z) }[/math]

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

[math]\displaystyle{ \operatorname{li}(e^x) = \operatorname{Ei}(x) }[/math]

for non-zero real values of [math]\displaystyle{ x }[/math].

Generalization

The exponential integral may also be generalized to

[math]\displaystyle{ E_n(x) = \int_1^\infty \frac{e^{-xt}}{t^n}\, dt, }[/math]

which can be written as a special case of the upper incomplete gamma function:[11]

[math]\displaystyle{ E_n(x) =x^{n-1}\Gamma(1-n,x). }[/math]

The generalized form is sometimes called the Misra function[12] [math]\displaystyle{ \varphi_m(x) }[/math], defined as

[math]\displaystyle{ \varphi_m(x)=E_{-m}(x). }[/math]

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[13]

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

The indefinite integral:

[math]\displaystyle{ \operatorname{Ei}(a \cdot b) = \iint e^{a b} \, da \, db }[/math]

is similar in form to the ordinary generating function for [math]\displaystyle{ d(n) }[/math], the number of divisors of [math]\displaystyle{ n }[/math]:

[math]\displaystyle{ \sum\limits_{n=1}^{\infty} d(n)x^{n} = \sum\limits_{a=1}^{\infty} \sum\limits_{b=1}^{\infty} x^{a b} }[/math]

Derivatives

The derivatives of the generalised functions [math]\displaystyle{ E_n }[/math] can be calculated by means of the formula [14]

[math]\displaystyle{ E_n '(z) = - E_{n-1}(z) \qquad (n=1,2,3,\ldots) }[/math]

Note that the function [math]\displaystyle{ E_0 }[/math] is easy to evaluate (making this recursion useful), since it is just [math]\displaystyle{ e^{-z}/z }[/math].[15]

Exponential integral of imaginary argument

[math]\displaystyle{ E_1(ix) }[/math] against [math]\displaystyle{ x }[/math]; real part black, imaginary part red.

If [math]\displaystyle{ z }[/math] is imaginary, it has a nonnegative real part, so we can use the formula

[math]\displaystyle{ E_1(z) = \int_1^\infty \frac{e^{-tz}} t \, dt }[/math]

to get a relation with the trigonometric integrals [math]\displaystyle{ \operatorname{Si} }[/math] and [math]\displaystyle{ \operatorname{Ci} }[/math]:

[math]\displaystyle{ E_1(ix) = i\left[ -\tfrac{1}{2}\pi + \operatorname{Si}(x)\right] - \operatorname{Ci}(x) \qquad (x \gt 0) }[/math]

The real and imaginary parts of [math]\displaystyle{ \mathrm{E}_1(ix) }[/math] 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[16] [math]\displaystyle{ E_1(x) = \left (A^{-7.7}+B \right )^{-0.13}, }[/math] where [math]\displaystyle{ \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} }[/math]
  • The Allen and Hastings approximation [16][17] [math]\displaystyle{ 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} }[/math] where [math]\displaystyle{ \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} }[/math]
  • The continued fraction expansion [17] [math]\displaystyle{ E_1(x) = \cfrac{e^{-x}}{x+\cfrac{1}{1+\cfrac{1}{x+\cfrac{2}{1+\cfrac{2}{x+\cfrac{3}{\ddots}}}}}}. }[/math]
  • The approximation of Barry et al. [18] [math]\displaystyle{ 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], }[/math] where: [math]\displaystyle{ \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} }[/math] with [math]\displaystyle{ \gamma }[/math] being the Euler–Mascheroni constant.

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[19]

See also

Notes

  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. ↑ Bleistein and Handelsman, p. 2
  7. ↑ Bleistein and Handelsman, p. 3
  8. ↑ O’Malley, Robert E. (2014), O'Malley, Robert E., ed., "Asymptotic Approximations" (in en), Historical Developments in Singular Perturbations (Cham: Springer International Publishing): pp. 27–51, doi:10.1007/978-3-319-11924-3_2, ISBN 978-3-319-11924-3, https://doi.org/10.1007/978-3-319-11924-3_2, retrieved 2023-05-04 
  9. ↑ Abramowitz and Stegun, p. 229, 5.1.20
  10. ↑ Abramowitz and Stegun, p. 228, see footnote 3.
  11. ↑ Abramowitz and Stegun, p. 230, 5.1.45
  12. ↑ After Misra (1940), p. 178
  13. ↑ Milgram (1985)
  14. ↑ Abramowitz and Stegun, p. 230, 5.1.26
  15. ↑ Abramowitz and Stegun, p. 229, 5.1.24
  16. ↑ 16.0 16.1 Giao, Pham Huy (2003-05-01). "Revisit of Well Function Approximation and An Easy Graphical Curve Matching Technique for Theis' Solution". Ground Water 41 (3): 387–390. doi:10.1111/j.1745-6584.2003.tb02608.x. ISSN 1745-6584. PMID 12772832. 
  17. ↑ 17.0 17.1 Tseng, Peng-Hsiang; Lee, Tien-Chang (1998-02-26). "Numerical evaluation of exponential integral: Theis well function approximation". Journal of Hydrology 205 (1–2): 38–51. doi:10.1016/S0022-1694(97)00134-0. Bibcode: 1998JHyd..205...38T. 
  18. ↑ Barry, D. A; Parlange, J. -Y; Li, L (2000-01-31). "Approximation for the exponential integral (Theis well function)". Journal of Hydrology 227 (1–4): 287–291. doi:10.1016/S0022-1694(99)00184-5. Bibcode: 2000JHyd..227..287B. 
  19. ↑ George I. Bell; Samuel Glasstone (1970). Nuclear Reactor Theory. Van Nostrand Reinhold Company. 

References

External links



Retrieved from "https://handwiki.org/wiki/index.php?title=Exponential_integral&oldid=3430266"