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

${\displaystyle E_{1}(z)=-\gamma -\ln z-\sum _{k=1}^{\infty }{\frac {(-z)^{k}}{k\;k!}}\qquad (|\operatorname {Arg} (z)|<\pi )}$

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

Semantic latex: \EulerpolyE{1}@{z} = - \EulerConstant - \ln z - \sum_{k=1}^{\infty} \frac{(-z)^k}{k k!} \qquad(\abs{\operatorname{Arg}(z)} < \cpi)

Confidence: 0.65121620744905

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(LaTeX -> Mathematica) Parenthesis mismatch in expression: Open and close parentheses does not match!

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(LaTeX -> SymPy) No translation possible for given token: Cannot extract information from feature set: \EulerpolyE [\EulerpolyE]

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(LaTeX -> Maple) Parenthesis mismatch in expression: Open and close parentheses does not match!

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$E_{1}(x)$

$\gamma$

$E_{1}$

$\mathrm {E} _{1}(ix)$

$\infty$

$E_{n}$

$z$

$E_{1}$

$E_{1}(z)=-\gamma -\ln z-\sum _{k=1}^{\infty }{\frac {(-z)^{k}}{k\;k!}}\qquad (|\operatorname {Arg} (z)|<\pi )$

$_{1}($

$E_{1}(z)$

Is part of

$E_{1}(z)=-\gamma -\ln z-\sum _{k=1}^{\infty }{\frac {(-z)^{k}}{k\;k!}}\qquad (|\operatorname {Arg} (z)|<\pi )$

Complete translation information:

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    "Maple" : {
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  "definiens" : [ ]
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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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