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| {{Distinguish2|other [[integral]]s of [[exponential function]]s}}
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| [[Image:Exponential integral.svg|300px|right|thumb| Plot of ''E''<sub>1</sub> function (top) and Ei function (bottom).]]
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| In mathematics, the '''exponential integral''' Ei is a [[special function]] on the [[complex plane]].
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| It is defined as one particular [[definite integral]] of the ratio between an [[exponential function]] and its [[argument of a function|argument]].
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| ==Definitions==
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| For real nonzero values of ''x'', the exponential integral Ei(''x'') is defined as
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| :<math> \operatorname{Ei}(x)=-\int_{-x}^{\infty}\frac{e^{-t}}t\,dt.\,</math>
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| 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.
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| For complex values of the argument, the definition becomes ambiguous due to [[branch points]] at 0 and <math>\infty</math>.<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>
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| :<math>\mathrm{E}_1(z) = \int_z^\infty \frac{e^{-t}}{t}\, dt,\qquad|{\rm Arg}(z)|<\pi</math>
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| In general, a [[branch cut]] is taken on the negative real axis and E<sub>1</sub> can be defined by [[analytic continuation]] elsewhere on the complex plane.
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| 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>
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| :<math>\mathrm{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>
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| The behaviour of E<sub>1</sub> near the branch cut can be seen by the following relation:<ref>Abramowitz and Stegun, p. 228, 5.1.7</ref>
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| :<math>\lim_{\delta\to0+}\mathrm{E_1}(-x \pm i\delta) = -\mathrm{Ei}(x) \mp i\pi,\qquad x>0,</math>
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| ==Properties==
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| Several properties of the exponential integral below, in certain cases, allow one to avoid its explicit evaluation through the definition above.
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| ===Convergent series===
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| Integrating the [[Taylor series]] for <math>e^{-t}/t</math>, and extracting the logarithmic singularity, we can derive the following series representation for <math>\mathrm{E_1}(x)</math> for real <math>x</math>:<ref>For a derivation, see Bender and Orszag, p253</ref>
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| :<math>\mathrm{Ei}(x) = \gamma+\ln |x| + \sum_{k=1}^{\infty} \frac{x^k}{k\; k!} \qquad x \neq 0</math>
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| For complex arguments off the negative real axis, this generalises to<ref>Abramowitz and Stegun, p. 229, 5.1.11</ref>
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| :<math>\mathrm{E_1}(z) =-\gamma-\ln z-\sum_{k=1}^{\infty}\frac{(-z)^k}{k\; k!} \qquad (|\mathrm{Arg}(z)| < \pi)</math>
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| 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.
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| This formula can be used to compute <math>\mathrm{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 [[loss of significance|cancellation]].
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| A faster converging series was found by [[Ramanujan]]:
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| :<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>
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| ===Asymptotic (divergent) series===
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| [[Image:AsymptoticExpansionE1.png|right|200px|thumb| Relative error of the asymptotic approximation for different number <math>~N~</math> of terms in the truncated sum]]
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| Unfortunately, the convergence of the series above is slow for arguments of larger modulus. For example, for x=10 more than 40 terms are required to get an answer correct to three significant figures.<ref>Bleistein and Handelsman, p. 2</ref> However, there is a divergent series approximation that can be obtained by integrating <math>ze^z\mathrm{E_1}(z)</math> by parts:<ref>Bleistein and Handelsman, p. 3</ref>
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| :<math>
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| \mathrm{E_1}(z)=\frac{\exp(-z)}{z}\sum_{n=0}^{N-1} \frac{n!}{(-z)^n}
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| </math>
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| which has error of order <math>O(N!z^{-N})</math> and is valid for large values of <math>\mathrm{Re}(z)</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).
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| ===Exponential and logarithmic behavior: bracketing===
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| [[Image:BracketingE1.png|right|200px|thumb|Bracketing of <math>\mathrm{E_1}</math> by elementary functions]]
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| From the two series suggested in previous subsections, it follows that <math>\mathrm{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>\mathrm{E_1}</math> can be bracketed by elementary functions as follows:<ref>Abramowitz and Stegun, p. 229, 5.1.20</ref>
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| :<math>
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| \frac{1}{2}e^{-x}\,\ln\!\left( 1+\frac{2}{x} \right)
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| < \mathrm{E_1}(x) <
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| e^{-x}\,\ln\!\left( 1+\frac{1}{x} \right)
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| \qquad x>0
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| </math>
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| The left-hand side of this inequality is shown in the graph to the left in blue; the central part <math>\mathrm{E_1}(x)</math> is shown in black and the right-hand side is shown in red.
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| ===Definition by <math>\mathrm{Ein}</math>===
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| Both <math>\mathrm{Ei}</math> and <math>\mathrm{E_1}</math> can be written more simply using the [[entire function]] <math>\mathrm{Ein}</math><ref>Abramowitz and Stegun, p. 228, see footnote 3.</ref> defined as
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| :<math>
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| \mathrm{Ein}(z)
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| = \int_0^z (1-e^{-t})\frac{dt}{t}
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| = \sum_{k=1}^\infty \frac{(-1)^{k+1}z^k}{k\; k!}
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| </math>
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| (note that this is just the alternating series in the above definition of <math>\mathrm{E_1}</math>). Then we have
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| :<math>
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| \mathrm{E_1}(z) \,=\, -\gamma-\ln z + {\rm Ein}(z)
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| \qquad |\mathrm{Arg}(z)| < \pi
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| </math>
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| :<math>\mathrm{Ei}(x) \,=\, \gamma+\ln x - \mathrm{Ein}(-x)
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| \qquad x>0
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| </math>
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| ===Relation with other functions===
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| The exponential integral is closely related to the [[logarithmic integral function]] li(''x'') by the formula
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| :<math>
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| \mathrm{li}(x) = \mathrm{Ei}(\ln x)\,
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| </math>
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| for positive real values of <math>x</math>
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| The exponential integral may also be generalized to
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| :<math>{\rm E}_n(x) = \int_1^\infty \frac{e^{-xt}}{t^n}\, dt,</math>
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| which can be written as a special case of the [[incomplete gamma function]]:<ref>Abramowitz and Stegun, p. 230, 5.1.45</ref>
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| : <math>{\rm E}_n(x) =x^{n-1}\Gamma(1-n,x).\,</math>
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| The generalized form is sometimes called the Misra function<ref>After Misra (1940), p. 178</ref> <math>\varphi_m(x)</math>, defined as
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| :<math>\varphi_m(x)={\rm E}_{-m}(x).\,</math>
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| Including a logarithm defines the generalized integro-exponential function<ref>Milgram (1985)</ref>
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| :<math>E_s^j(z)= \frac{1}{\Gamma(j+1)}\int_1^\infty (\log t)^j \frac{e^{-zt}}{t^s}\,dt</math>.
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| The indefinite integral:
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| :<math> \mathrm{Ei}(a \cdot b) = \iint e^{a b} \, da \, db</math>
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| is similar in form to the ordinary [[generating function]] for <math>d(n)</math>, the number of [[divisors]] of <math>n</math>:
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| :<math> \sum\limits_{n=1}^{\infty} d(n)x^{n} = \sum\limits_{a=1}^{\infty} \sum\limits_{b=1}^{\infty} x^{a b}</math>
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| ===Derivatives===
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| The derivatives of the generalised functions <math>\mathrm{E_n}</math> can be calculated by means of the formula <ref>Abramowitz and Stegun, p. 230, 5.1.26</ref>
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| :<math>
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| \mathrm{E_n}'(z) = -\mathrm{E_{n-1}}(z)
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| \qquad (n=1,2,3,\ldots)
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| </math>
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| Note that the function <math>\mathrm{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>
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| ===Exponential integral of imaginary argument===
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| [[Image:E1ofImaginaryArgument.png|right|200px|thumb|<math>\mathrm{E_1}(ix)</math>
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| against <math>x</math>; real part black, imaginary part red.]]
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| If <math>z</math> is imaginary, it has a nonnegative real part, so we can use the formula
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| :<math>
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| \mathrm{E_1}(z) = \int_1^\infty
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| \frac{e^{-tz}}{t} dt
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| </math>
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| to get a relation with the [[trigonometric integrals]] <math>\mathrm{Si}</math> and <math>\mathrm{Ci}</math>:
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| :<math> | |
| \mathrm{E_1}(ix) = i\left(-\tfrac{1}{2}\pi + \mathrm{Si}(x)\right) - \mathrm{Ci}(x)
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| \qquad (x>0)
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| </math>
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| The real and imaginary parts of <math>\mathrm{E_1}(x)</math> are plotted in the figure to the right with black and red curves.
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| == Applications ==
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| * Time-dependent [[heat transfer]]
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| * Nonequilibrium [[groundwater]] flow in the [[Aquifer test#Transient Theis solution|Theis solution]] (called a ''well function'')
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| * Radiative transfer in stellar atmospheres
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| * Radial diffusivity equation for transient or unsteady state flow with line sources and sinks
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| * Solutions to the [[neutron transport]] equation in simplified 1-D geometries.<ref>{{cite book|title=Nuclear Reactor Theory|year=1970|publisher=Van Nostrand Reinhold Company|author=George I. Bell|coauthors=Samuel Glasstone}}</ref>
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| ==Notes==
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| {{reflist|2}}
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| ==References==
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| * {{cite book
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| | last = Abramovitz
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| | first = Milton
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| | others= [[Abramowitz and Stegun]]
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| | coauthors =Irene Stegun
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| | title =Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables
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| | publisher = Dover
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| | year = 1964
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| | location = New York
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| | url = http://www.math.sfu.ca/~cbm/aands
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| | isbn = 0-486-61272-4 }}, [http://people.math.sfu.ca/~cbm/aands/page_228.htm Chapter 5].
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| * {{cite book
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| | last = Bender
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| | first = Carl M.
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| | coauthors = Steven A. Orszag
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| | title = Advanced mathematical methods for scientists and engineers
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| | publisher = McGraw–Hill
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| | year = 1978
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| | isbn = 0-07-004452-X
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| }}
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| * {{cite book
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| | last = Bleistein
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| | first = Norman
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| | coauthors = Richard A. Handelsman
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| | title = Asymptotic Expansions of Integrals
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| | publisher = Dover
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| | year = 1986
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| | isbn = 0-486-65082-0
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| }}
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| * {{cite journal
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| |doi=10.1093/qmath/1.1.176
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| |first=Ida W.
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| |last=Busbridge
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| |journal=Quart. J. Math. (Oxford)
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| |year=1950
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| |volume=1
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| |issue=1
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| |title=On the integro-exponential function and the evaluation of some integrals involving it
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| |pages=176–184
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| |bibcode=1950QJMat...1..176B
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| }}
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| * {{cite journal
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| |first1=A.
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| |last1=Stankiewicz
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| |title=Tables of the integro-exponential functions
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| |journal=Acta Astronomica
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| |volume=18
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| |page=289
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| |year=1968
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| |bibcode=1968AcA....18..289S
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| }}
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| * {{cite journal
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| |first1=R. R.
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| |last1=Sharma
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| |first2=Bahman
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| |last2=Zohuri
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| |title=A general method for an accurate evaluation of exponential integrals E<sub>1</sub>(x), x>0
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| |journal=J. Comput. Phys.
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| |volume=25
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| |number=2
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| |pages=199—204
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| |doi=10.1016/0021-9991(77)90022-5
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| |year=1977
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| |bibcode=1977JCoPh..25..199S
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| }}
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| * {{cite journal
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| |doi=10.1090/S0025-5718-1983-0701632-1
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| |first1=K. S.
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| |last1=Kölbig
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| |title=On the integral exp(−''μt'')''t''<sup>ν−1</sup>log<sup>''m''</sup>''t'' ''dt''
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| |journal=Math. Comput
| |
| |year=1983
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| |pages=171—182
| |
| |volume=41
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| |number=163
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| }}
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| * {{cite journal
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| |doi=10.1090/S0025-5718-1985-0777276-4
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| |first=M. S.
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| |last=Milgram
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| |journal=Mathematics of Computation
| |
| |title=The generalized integro-exponential function
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| |volume=44
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| |issue=170
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| |year=1985
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| |mr=0777276
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| |pages=443–458
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| |jstor = 2007964
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| }}
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| * {{cite journal
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| | last = Misra
| |
| | first = Rama Dhar
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| | year = 1940
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| | title = On the Stability of Crystal Lattices. II
| |
| | journal = Mathematical Proceedings of the Cambridge Philosophical Society
| |
| | volume = 36
| |
| | issue = 2
| |
| | pages = 173
| |
| | doi = 10.1017/S030500410001714X
| |
| | last2 = Born
| |
| | first2 = M.
| |
| |bibcode = 1940PCPS...36..173M }}
| |
| * {{cite journal
| |
| |first1=C.
| |
| |last1=Chiccoli
| |
| |first2=S.
| |
| |last2=Lorenzutta
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| |first3=G.
| |
| |last3=Maino
| |
| |title=On the evaluation of generalized exponential integrals E<sub>ν</sub>(x)
| |
| |journal=J. Comput. Phys.
| |
| |volume=78
| |
| |pages=278—287
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| |year=1988
| |
| |doi=10.1016/0021-9991(88)90050-2
| |
| |bibcode=1988JCoPh..78..278C
| |
| }}
| |
| * {{cite journal
| |
| |first1=C.
| |
| |last1=Chiccoli
| |
| |first2=S.
| |
| |last2=Lorenzutta
| |
| |first3=G.
| |
| |last3=Maino
| |
| |title=Recent results for generalized exponential integrals
| |
| |journal=Computer Math. Applic.
| |
| |volume=19
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| |number=5
| |
| |pages=21—29
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| |year=1990
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| |doi=10.1016/0898-1221(90)90098-5
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| }}
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| * {{cite journal
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| |first1=Allan J.
| |
| |last1=MacLeod
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| |title=The efficient computation of some generalised exponential integrals
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| |journal=J. Comput. Appl. Math.
| |
| |doi=10.1016/S0377-0427(02)00556-3
| |
| |year=2002
| |
| |volume=148
| |
| |number=2
| |
| |pages=363—374
| |
| |bibcode=2002JCoAm.138..363M
| |
| }}
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| * {{Citation | last1=Press | first1=WH | last2=Teukolsky | first2=SA | last3=Vetterling | first3=WT | last4=Flannery | first4=BP | year=2007 | title=Numerical Recipes: The Art of Scientific Computing | edition=3rd | publisher=Cambridge University Press | publication-place=New York | isbn=978-0-521-88068-8 | chapter=Section 6.3. Exponential Integrals | chapter-url=http://apps.nrbook.com/empanel/index.html#pg=266}}
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| *{{dlmf|id=6|title=Exponential, Logarithmic, Sine, and Cosine Integrals|first=N. M. |last=Temme}}
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| == External links ==
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| * {{springer|title=Integral exponential function|id=p/i051440}}
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| * [http://dlmf.nist.gov/8.19 NIST documentation on the Generalized Exponential Integral]
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| *{{MathWorld|urlname=ExponentialIntegral|title=Exponential Integral}}
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| *{{MathWorld|urlname=En-Function|title=''En''-Function}}
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| * [http://functions.wolfram.com/GammaBetaErf/ExpIntegralEi/ Formulas and identities for Ei]
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| {{DEFAULTSORT:Exponential Integral}}
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| [[Category:Exponentials]]
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| [[Category:Special functions]]
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| [[Category:Special hypergeometric functions]]
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| [[Category:Integrals]]
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