Editing Exponential integral (section)

=== 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]].