LaTeX to CAS translator

This mockup demonstrates the concept of TeX to Computer Algebra System (CAS) conversion.

The demo-application converts LaTeX functions which directly translate to CAS counterparts.

Functions without explicit CAS support are available for translation via a DRMF package (under development).

The following LaTeX input ...

{\displaystyle \operatorname{Ei}}

${\displaystyle \operatorname {Ei} }$

... is translated to the CAS output ...

Semantic latex: \operatorname{Ei}

Confidence: 0

Mathematica

Translation: E*i

Information

Sub Equations

E*i

Free variables

E

i

Symbol info

Was interpreted as a function call because of a leading \operatorname.

Tests

Symbolic

Numeric

SymPy

Translation: E*i

Information

Sub Equations

E*i

Free variables

E

i

Symbol info

Was interpreted as a function call because of a leading \operatorname.

Tests

Symbolic

Numeric

Maple

Translation: E*i

Information

Sub Equations

E*i

Free variables

E

i

Symbol info

Was interpreted as a function call because of a leading \operatorname.

Tests

Symbolic

Numeric

Dependency Graph Information

Is part of

$-E_{1}(x)=\operatorname {Ei} (-x)$

${\rm {Ei}}(x)=\gamma +\ln x+\exp {(x/2)}\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}x^{n}}{n!\,2^{n-1}}}\sum _{k=0}^{\lfloor (n-1)/2\rfloor }{\frac {1}{2k+1}}$

$1-{\frac {3x}{4}}\leq {\rm {Ei}}(x)-\gamma -\ln x\leq 1-{\frac {3x}{4}}+{\frac {11x^{2}}{36}}$

$\lim _{\delta \to 0+}E_{1}(-x\pm i\delta )=-\operatorname {Ei} (x)\mp i\pi ,\qquad x>0$

$\operatorname {Ei} (x)\,=\,\gamma +\ln x-\operatorname {Ein} (-x)\qquad x>0$

$\operatorname {li} (e^{x})=\operatorname {Ei} (x)$

$\operatorname {Ei} (a\cdot b)=\iint e^{ab}\,da\,db$

Description

series in the above definition

entire function

note

series

formula

non-zero real value

exponential time

function

logarithmic integral function li

good asymptotic bound for small x

Ramanujan

form to the ordinary generating function

indefinite integral

number of divisor

behaviour

relation

positive value

exponential integral

e.g.

connexion with the confluent hypergeometric function

branch cut

ei

notation

Complete translation information:

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Specify your own input

Formula input
Wikitext context	{{Use American English\|date = January 2019}} {{Short description\|Special function defined by an integral}} {{distinguish\|text=[[List of integrals of exponential functions\|other integrals]] of [[exponential function]]s}} [[Image:Exponential integral.svg\|300px\|right\|thumb\| Plot of <math>E_1</math> function (top) and <math>\operatorname{Ei}</math> function (bottom).]] 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 of a function\|argument]]. ==Definitions== 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 <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> :<math>E_1(z) = \int_z^\infty \frac{e^{-t}}{t}\, dt,\qquad\|{\rm Arg}(z)\|<\pi</math> (note that for positive values of  ''x'', we have <math>-E_1(x) = \operatorname{Ei}(-x)</math>). 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. 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 ''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> :<math>\lim_{\delta\to0+} E_1(-x \pm i\delta) = -\operatorname{Ei}(x) \mp i\pi,\qquad x>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=== 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 (\|\operatorname{Arg}(z)\| < \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 [[loss of significance\|cancellation]]. A faster converging series was found by [[Ramanujan]]: :<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> These alternating series can also be used to give good asymptotic bounds for small x, e.g. {{Citation needed\|reason=upper bound appears to be incorrect\|date=June 2020}}: :<math>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>x\ge 0</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, for ''x'' = 10 more than 40 terms are required to get an answer correct to three significant figures for <math>E_1(z)</math>.<ref>Bleistein and Handelsman, p. 2</ref> However, there is a divergent series approximation that can be obtained by integrating <math>z e^z E_1(z)</math> by parts:<ref>Bleistein and Handelsman, p. 3</ref> : <math>E_1(z)=\frac{\exp(-z)} z \sum_{n=0}^{N-1} \frac{n!}{(-z)^n} </math> which has error of order <math>O(N!z^{-N})</math> and is valid for large values of <math>\operatorname{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). ===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. 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. 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>\mathrm{E}_1</math>). Then we have :<math> E_1(z) \,=\, -\gamma-\ln z + {\rm Ein}(z) \qquad \|\operatorname{Arg}(z)\| < \pi </math> :<math>\operatorname{Ei}(x) \,=\, \gamma+\ln x - \operatorname{Ein}(-x) \qquad x>0 </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>. 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 [[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> 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 (\log t)^j \frac{e^{-zt}}{t^s}\,dt.</math> The indefinite integral: :<math> \operatorname{Ei}(a \cdot b) = \iint e^{a b} \, da \, db</math> is similar in form to the ordinary [[generating function]] for <math>d(n)</math>, the number of [[divisors]] of <math>n</math>: :<math> \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>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 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}}</ref> :: <math>E_1(x) = \left (A^{-7.7}+B \right )^{-0.13},</math> :where ::<math>\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\|first = Peng-Hsiang\|last = Tseng\|first2 = Tien-Chang\|last2 = Lee\|bibcode = 1998JHyd..205...38T }}</ref> ::<math>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>\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>E_1(x) = \cfrac{e^{-x}}{x+\cfrac{1}{1+\cfrac{1}{x+\cfrac{2}{1+\cfrac{2}{x+\cfrac{3}{\dots}}}}}}.</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\|first = D. A\|last = Barry\|first2 = J. -Y\|last2 = Parlange \|first3 = L\|last3 = Li\|bibcode = 2000JHyd..227..287B }}</ref> ::<math>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>\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]]. == Applications == * Time-dependent [[heat transfer]] * Nonequilibrium [[groundwater]] flow in the [[Aquifer test#Transient Theis solution\|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>{{cite book\|title=Nuclear Reactor Theory\|year=1970\|publisher=Van Nostrand Reinhold Company\|author=George I. Bell\|author2=Samuel Glasstone}}</ref> ==See also== * [[Goodwin–Staton integral]] * [[Bickley–Naylor functions]] ==Notes== {{reflist\|2}} ==References== * {{cite book \|last = Abramowitz \|first = Milton \|others = [[Abramowitz and Stegun]] \|author2 = Irene Stegun \|title = Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables \|publisher = Dover \|year = 1964 \|location = New York \|url = https://archive.org/details/handbookofmathe000abra \|isbn = 978-0-486-61272-0 \|url-access = registration }}, [http://people.math.sfu.ca/~cbm/aands/page_228.htm Chapter 5]. * {{cite book \| last = Bender \| first = Carl M. \|author2=Steven A. Orszag \| title = Advanced mathematical methods for scientists and engineers \| publisher = McGraw–Hill \| year = 1978 \| isbn = 978-0-07-004452-4 }} * {{cite book \| last = Bleistein \| first = Norman \|author2=Richard A. Handelsman \| title = Asymptotic Expansions of Integrals \| publisher = Dover \| year = 1986 \| isbn = 978-0-486-65082-1 }} * {{cite journal \|doi=10.1093/qmath/1.1.176 \|first=Ida W. \|last=Busbridge \|journal=Quart. J. Math. (Oxford) \|year=1950 \|volume=1 \|issue=1 \|title=On the integro-exponential function and the evaluation of some integrals involving it \|pages=176–184 \|bibcode=1950QJMat...1..176B }} * {{cite journal \|first1=A. \|last1=Stankiewicz \|title=Tables of the integro-exponential functions \|journal=Acta Astronomica \|volume=18 \|page=289 \|year=1968 \|bibcode=1968AcA....18..289S }} * {{cite journal \|first1=R. R. \|last1=Sharma \|first2=Bahman \|last2=Zohuri \|title=A general method for an accurate evaluation of exponential integrals E<sub>1</sub>(x), x>0 \|journal=J. Comput. Phys. \|volume=25 \|number=2 \|pages=199–204 \|doi=10.1016/0021-9991(77)90022-5 \|year=1977 \|bibcode=1977JCoPh..25..199S }} * {{cite journal \|doi=10.1090/S0025-5718-1983-0701632-1 \|first1=K. S. \|last1=Kölbig \|title=On the integral exp(−''μt'')''t''<sup>ν−1</sup>log<sup>''m''</sup>''t'' ''dt'' \|journal=Math. Comput. \|year=1983 \|pages=171–182 \|volume=41 \|number=163 \|doi-access=free }} * {{cite journal \|doi=10.1090/S0025-5718-1985-0777276-4 \|first=M. S. \|last=Milgram \|journal=Mathematics of Computation \|title=The generalized integro-exponential function \|volume=44 \|issue=170 \|year=1985 \|mr=0777276 \|pages=443–458 \|jstor = 2007964 \|doi-access=free }} * {{cite journal \| last = Misra \| first = Rama Dhar \| year = 1940 \| 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 \|first3=G. \|last3=Maino \|title=On the evaluation of generalized exponential integrals E<sub>ν</sub>(x) \|journal=J. Comput. Phys. \|volume=78 \|issue=2 \|pages=278–287 \|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 \|number=5 \|pages=21–29 \|year=1990 \|doi=10.1016/0898-1221(90)90098-5 }} * {{cite journal \|first1=Allan J. \|last1=MacLeod \|title=The efficient computation of some generalised exponential integrals \|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 \|doi-access=free }} * {{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 \| location=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}} {{dlmf\|id=6\|title=Exponential, Logarithmic, Sine, and Cosine Integrals\|first=N. M. \|last=Temme}} == External links == {{springer\|title=Integral exponential function\|id=p/i051440}} * [http://dlmf.nist.gov/8.19 NIST documentation on the Generalized Exponential Integral] {{MathWorld\|urlname=ExponentialIntegral\|title=Exponential Integral}} {{MathWorld\|urlname=En-Function\|title=''En''-Function}} * {{WolframFunctionsSite \| urlname=GammaBetaErf/ExpIntegralEi/ \| title= Exponential integral Ei}} * [http://dlmf.nist.gov/6 Exponential, Logarithmic, Sine, and Cosine Integrals] in [[DLMF]]. <!-- {{DLMF}} is not for external links. ;-/ --> {{Nonelementary Integral}} {{DEFAULTSORT:Exponential Integral}} [[Category:Exponentials]] [[Category:Special functions]] [[Category:Special hypergeometric functions]] [[Category:Integrals]]
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