Editing Exponential integral (section)

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

===Convergent series===
[[Image:Exponential integral.svg|300px|right|thumb| 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.&nbsp;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 [[Catastrophic cancellation|cancellation]].

A faster converging series was found by [[Ramanujan]]:<ref>Andrews and Berndt, p.&nbsp;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) series===
[[Image:AsymptoticExpansionE1.png|right|200px|thumb| 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.&nbsp;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.&nbsp;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 orders ====
Using integration by parts, we can obtain an explicit formula<ref>{{Citation |last=O’Malley |first=Robert E. |title=Asymptotic Approximations |date=2014 |url=https://doi.org/10.1007/978-3-319-11924-3_2 |work=Historical Developments in Singular Perturbations |pages=27–51 |editor-last=O'Malley |editor-first=Robert E. |access-date=2023-05-04 |place=Cham |publisher=Springer International Publishing |language=en |doi=10.1007/978-3-319-11924-3_2 |isbn=978-3-319-11924-3|url-access=subscription }}</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: bracketing===
[[Image:BracketingE1.png|right|200px|thumb|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.&nbsp;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 Ein===

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.&nbsp;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 functions===
Kummer's equation
:<math>z\frac{d^2w}{dz^2} + (b-z)\frac{dw}{dz} - aw = 0</math>
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,
:<math>z\frac{d^2w}{dz^2} + (1-z)\frac{dw}{dz} = 0</math>
we have
:<math>M(0,1,z)=U(0,1,z)=1</math>
for all ''z''. A second solution is then given by E<sub>1</sub>(−''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 ''E<sub>1</sub>'' is an exponential times the function ''U''(1,1,''z''):
:<math>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>\operatorname{li}(e^x) = \operatorname{Ei}(x)</math>
for non-zero real values of <math>x </math>.

===Generalization===
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.&nbsp;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.&nbsp;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 [https://dlmf.nist.gov/8.19 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>

===Derivatives===

The derivatives of the generalised functions <math>E_n</math> can be calculated by means of the formula <ref>Abramowitz and Stegun, p.&nbsp;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.&nbsp;229, 5.1.24</ref>

===Exponential integral of imaginary argument===

[[Image:E1ofImaginaryArgument.png|right|200px|thumb|<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 integral]]s <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.

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

* The Swamee and Ohija approximation<ref name=":0">{{Cite journal|title = Revisit of Well Function Approximation and An Easy Graphical Curve Matching Technique for Theis' Solution|journal = Ground Water|date = 2003-05-01|issn = 1745-6584|pages = 387–390|volume = 41| issue = 3|doi = 10.1111/j.1745-6584.2003.tb02608.x|first = Pham Huy|last = Giao| pmid=12772832 | bibcode=2003GrWat..41..387G | s2cid=31982931 }}</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">{{Cite journal|title = Numerical evaluation of exponential integral: Theis well function approximation|journal = Journal of Hydrology|date = 1998-02-26|pages = 38–51|volume = 205|issue = 1–2|doi = 10.1016/S0022-1694(97)00134-0|first1 = Peng-Hsiang|last1 = Tseng|first2 = Tien-Chang|last2 = Lee|bibcode = 1998JHyd..205...38T }}</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>{{Cite journal|title = Approximation for the exponential integral (Theis well function) |journal = Journal of Hydrology|date = 2000-01-31| pages = 287–291|volume = 227|issue = 1–4|doi = 10.1016/S0022-1694(99)00184-5|first1 = D. A|last1 = Barry|first2 = J. -Y|last2 = Parlange |first3 = L|last3 = Li|bibcode = 2000JHyd..227..287B }}</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]].