Exponential integral

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File: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.svg

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.

DefinitionsEdit

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

<math> \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 Template:Nowrap<ref>Abramowitz and Stegun, p. 228</ref> Instead of Ei, the following notation is used,<ref>Abramowitz and Stegun, p. 228, 5.1.1</ref>

<math>E_1(z) = \int_z^\infty \frac{e^{-t}}{t}\, dt,\qquad|{\rm Arg}(z)|<\pi</math>

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

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 Template:Nowrap

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>z</math>, this can be written<ref>Abramowitz and Stegun, p. 228, 5.1.4 with n = 1</ref>

<math>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:<ref>Abramowitz and Stegun, p. 228, 5.1.7</ref>

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

PropertiesEdit

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

Convergent seriesEdit

File:Exponential integral.svg

Plot of <math>E_1</math> function (top) and <math>\operatorname{Ei}</math> function (bottom).

For real or complex arguments off the negative real axis, <math>E_1(z)</math> can be expressed as<ref>Abramowitz and Stegun, p. 229, 5.1.11</ref>

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

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

A faster converging series was found by Ramanujan:<ref>Andrews and Berndt, p. 130, 24.16</ref>

<math>{\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>

Asymptotic (divergent) seriesEdit

File:AsymptoticExpansionE1.png

Relative error of the asymptotic approximation for different number <math>~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>E_1(10)</math>.<ref>Bleistein and Handelsman, p. 2</ref> However, for positive values of x, there is a divergent series approximation that can be obtained by integrating <math>x e^x E_1(x)</math> by parts:<ref>Bleistein and Handelsman, p. 3</ref>

<math>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>N</math>, the number of terms in the truncated sum (<math>N=1</math> in red, <math>N=5</math> in pink).

Asymptotics beyond all ordersEdit

Using integration by parts, we can obtain an explicit formula<ref>Template:Citation</ref><math display="block">\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>z</math>, the absolute value of the error term <math>|e_n(z)|</math> decreases, then increases. The minimum occurs at <math>n\sim |z|</math>, at which point <math>\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: bracketingEdit

File:BracketingE1.png

Bracketing of <math>E_1</math> by elementary functions

From the two series suggested in previous subsections, it follows that <math>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>E_1</math> can be bracketed by elementary functions as follows:<ref>Abramowitz and Stegun, p. 229, 5.1.20</ref>

<math>

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

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

Definition by EinEdit

Both <math>\operatorname{Ei}</math> and <math>E_1</math> can be written more simply using the entire function <math>\operatorname{Ein}</math><ref>Abramowitz and Stegun, p. 228, see footnote 3.</ref> defined as

<math>

\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>E_1</math>). Then we have

<math>

E_1(z) \,=\, -\gamma-\ln z + {\rm Ein}(z) \qquad \left| \operatorname{Arg}(z) \right| < \pi </math>

<math>\operatorname{Ei}(x) \,=\, \gamma+\ln{x} - \operatorname{Ein}(-x)

\qquad x \neq 0 </math> The function <math>\operatorname{Ein}</math> is related to the exponential generating function of the harmonic numbers:

<math>

\operatorname{Ein}(z) = e^{-z} \, \sum_{n=1}^\infty \frac {z^n}{n!} H_n </math>

Relation with other functionsEdit

Kummer's equation

is usually solved by the confluent hypergeometric functions <math>M(a,b,z)</math> and <math>U(a,b,z).</math> But when <math>a=0</math> and <math>b=1,</math> that is,

we have

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

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

with the derivative evaluated at <math>a=0.</math> Another connexion with the confluent hypergeometric functions is that E₁ is an exponential times the function U(1,1,z):

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

<math>\operatorname{li}(e^x) = \operatorname{Ei}(x)</math>

for non-zero real values of <math>x </math>.

GeneralizationEdit

The exponential integral may also be generalized to

<math>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:<ref>Abramowitz and Stegun, p. 230, 5.1.45</ref>

<math>E_n(x) =x^{n-1}\Gamma(1-n,x).</math>

The generalized form is sometimes called the Misra function<ref>After Misra (1940), p. 178</ref> <math>\varphi_m(x)</math>, defined as

<math>\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<ref>Milgram (1985)</ref>

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

DerivativesEdit

The derivatives of the generalised functions <math>E_n</math> can be calculated by means of the formula <ref>Abramowitz and Stegun, p. 230, 5.1.26</ref>

<math>

E_n '(z) = - E_{n-1}(z) \qquad (n=1,2,3,\ldots) </math> Note that the function <math>E_0</math> is easy to evaluate (making this recursion useful), since it is just <math>e^{-z}/z</math>.<ref>Abramowitz and Stegun, p. 229, 5.1.24</ref>

Exponential integral of imaginary argumentEdit

File:E1ofImaginaryArgument.png

<math>E_1(ix)</math> against <math>x</math>; real part black, imaginary part red.

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

<math>

E_1(z) = \int_1^\infty \frac{e^{-tz}} t \, dt </math> to get a relation with the trigonometric integrals <math>\operatorname{Si}</math> and <math>\operatorname{Ci}</math>:

<math>

E_1(ix) = i\left[ -\tfrac{1}{2}\pi + \operatorname{Si}(x)\right] - \operatorname{Ci}(x) \qquad (x > 0) </math> The real and imaginary parts of <math>\mathrm{E}_1(ix)</math> are plotted in the figure to the right with black and red curves.

ApproximationsEdit

There have been a number of approximations for the exponential integral function. These include:

The Swamee and Ohija approximation<ref name=":0">Template:Cite journal</ref> <math display="block">E_1(x) = \left (A^{-7.7}+B \right )^{-0.13},</math> where <math display="block">\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 <ref name=":0" /><ref name=":1">Template:Cite journal</ref> <math display="block">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 display="block">\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 <ref name=":1" /> <math display="block">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. <ref>Template:Cite journal</ref> <math display="block">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 display="block">\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>\gamma</math> being the Euler–Mascheroni constant.

Inverse function of the Exponential IntegralEdit

We can express the Inverse function of the exponential integral in power series form:<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

<math>\forall |x| < \frac{\mu}{\ln(\mu)},\quad \mathrm{Ei}^{-1}(x) = \sum_{n=0}^\infty \frac{x^n}{n!} \frac{P_n(\ln(\mu))}{\mu^n}</math>

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

For <math>n > 0</math>, <math>\deg P_n = n</math> and we have the formula :

<math>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}.</math>

ApplicationsEdit

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<ref>Template:Cite book</ref>
Solutions to the Trachenko-Zaccone nonlinear differential equation for the stretched exponential function in the relaxation of amorphous solids and glass transition<ref>Template:Cite journal</ref><ref>Template:Cite journal</ref>