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 O}

${\displaystyle O}$

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

Semantic latex: O

Confidence: 0

Mathematica

Translation: O

Information

Sub Equations

O

Free variables

O

Tests

Symbolic

Numeric

SymPy

Translation: O

Information

Sub Equations

O

Free variables

O

Tests

Symbolic

Numeric

Maple

Translation: O

Information

Sub Equations

O

Free variables

O

Tests

Symbolic

Numeric

Dependency Graph Information

Is part of

$\operatorname {li} (x)=O\left({x \over \ln x}\right)\;$

$\operatorname {li} (x)-{x \over \ln x}=O\left({x \over \ln ^{2}x}\right)\;$

$\operatorname {Li} (x)-\pi (x)=O({\sqrt {x}}\log x)$

Description

big O notation

asymptotic behavior

accurate asymptotic behaviour

Riemann hypothesis

Complete translation information:

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    "SymPy" : {
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      },
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  "positions" : [ {
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    "sentence" : 1,
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  "includes" : [ ],
  "isPartOf" : [ "\\operatorname{li}(x) = O \\left( {x\\over \\ln x} \\right) \\;", "\\operatorname{li}(x) - {x\\over \\ln x} = O \\left( {x\\over \\ln^2 x} \\right) \\;", "\\operatorname{Li}(x)-\\pi(x) = O(\\sqrt{x}\\log x)" ],
  "definiens" : [ {
    "definition" : "big O notation",
    "score" : 0.722
  }, {
    "definition" : "asymptotic behavior",
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  }, {
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  }, {
    "definition" : "Riemann hypothesis",
    "score" : 0.30293665845946705
  } ]
}

Specify your own input

Formula input
Wikitext context	{{Use American English\|date = January 2019}} {{Redirect\|Li(x)\|the polylogarithm denoted by Li<sub>''s''</sub>(''z'')\|Polylogarithm}} {{Short description\|Special function defined by an integral}} In [[mathematics]], the '''logarithmic integral function''' or '''integral logarithm''' li(''x'') is a [[special function]]. It is relevant in problems of [[physics]] and has [[number theory\|number theoretic]] significance. In particular, according to the [[Siegel-Walfisz theorem]] it is a very good [[approximation]] to the [[prime-counting function]], which is defined as the number of [[prime numbers]] less than or equal to a given value <math>x</math>. [[Image:Logarithmic integral function.svg\|thumb\|right\|300px\|Logarithmic integral function plot]] ==Integral representation== The logarithmic integral has an integral representation defined for all positive [[real number]]s {{mvar\|x}} ≠ 1 by the [[integral\|definite integral]] :<math> \operatorname{li}(x) = \int_0^x \frac{dt}{\ln t}. </math> Here, {{math\|ln}} denotes the [[natural logarithm]]. The function {{math\|1/(ln ''t'')}} has a [[mathematical singularity\|singularity]] at {{math\|1=''t'' = 1}}, and the integral for {{math\|''x'' > 1}} is interpreted as a [[Cauchy principal value]], :<math> \operatorname{li}(x) = \lim_{\varepsilon \to 0+} \left( \int_0^{1-\varepsilon} \frac{dt}{\ln t} + \int_{1+\varepsilon}^x \frac{dt}{\ln t} \right).</math> ==Offset logarithmic integral== The '''offset logarithmic integral''' or '''Eulerian logarithmic integral''' is defined as :<math> \operatorname{Li}(x) = \int_2^x \frac{dt}{\ln t} = \operatorname{li}(x) - \operatorname{li}(2). </math> As such, the integral representation has the advantage of avoiding the singularity in the domain of integration. ==Special values== The function li(''x'') has a single positive zero; it occurs at ''x'' ≈ 1.45136 92348 83381 05028 39684 85892 02744 94930... {{OEIS2C\|A070769}}; this number is known as the [[Ramanujan–Soldner constant]]. −Li(0) = li(2) ≈ 1.045163 780117 492784 844588 889194 613136 522615 578151... {{OEIS2C\|A069284}} This is <math>-(\Gamma\left(0,-\ln 2\right) + i\,\pi)</math> where <math>\Gamma\left(a,x\right)</math> is the [[incomplete gamma function]]. It must be understood as the [[Cauchy principal value]] of the function. ==Series representation== The function li(''x'') is related to the ''[[exponential integral]]'' Ei(''x'') via the equation :<math>\hbox{li}(x)=\hbox{Ei}(\ln x) , \,\!</math> which is valid for ''x'' > 0. This identity provides a series representation of li(''x'') as :<math> \operatorname{li}(e^u) = \hbox{Ei}(u) = \gamma + \ln \|u\| + \sum_{n=1}^\infty {u^{n}\over n \cdot n!} \quad \text{ for } u \ne 0 \; , </math> where γ ≈ 0.57721 56649 01532 ... {{OEIS2C\|id=A001620}} is the [[Euler–Mascheroni constant]]. A more rapidly convergent series by [[Srinivasa Ramanujan\|Ramanujan]] <ref>{{MathWorld \| urlname=LogarithmicIntegral \| title=Logarithmic Integral}}</ref> is :<math> \operatorname{li}(x) = \gamma + \ln \ln x + \sqrt{x} \sum_{n=1}^\infty \frac{ (-1)^{n-1} (\ln x)^n} {n! \, 2^{n-1}} \sum_{k=0}^{\lfloor (n-1)/2 \rfloor} \frac{1}{2k+1} . </math> <!-- cribbed from Mathworld, which cites Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, pp. 126–131, 1994. --> ==Asymptotic expansion== The asymptotic behavior for ''x'' → ∞ is :<math> \operatorname{li}(x) = O \left( {x\over \ln x} \right) \; . </math> where <math>O</math> is the [[big O notation]]. The full [[asymptotic expansion]] is :<math> \operatorname{li}(x) \sim \frac{x}{\ln x} \sum_{k=0}^\infty \frac{k!}{(\ln x)^k} </math> or :<math> \frac{\operatorname{li}(x)}{x/\ln x} \sim 1 + \frac{1}{\ln x} + \frac{2}{(\ln x)^2} + \frac{6}{(\ln x)^3} + \cdots. </math> This gives the following more accurate asymptotic behaviour: :<math> \operatorname{li}(x) - {x\over \ln x} = O \left( {x\over \ln^2 x} \right) \; . </math> As an asymptotic expansion, this series is [[divergent series\|not convergent]]: it is a reasonable approximation only if the series is truncated at a finite number of terms, and only large values of ''x'' are employed. This expansion follows directly from the asymptotic expansion for the [[exponential integral]]. This implies e.g. that we can bracket li as: :<math> 1+\frac{1}{\ln x} < \operatorname{li}(x) \frac{\ln x}{x} < 1+\frac{1}{\ln x}+\frac{3}{(\ln x)^2} </math> for all <math>\ln x \ge 11</math>. ==Number theoretic significance== The logarithmic integral is important in [[number theory]], appearing in estimates of the number of [[prime number]]s less than a given value. For example, the [[prime number theorem]] states that: :<math>\pi(x)\sim\operatorname{Li}(x)</math> where <math>\pi(x)</math> denotes the number of primes smaller than or equal to <math>x</math>. Assuming the [[Riemann hypothesis]], we get the even stronger:<ref>Abramowitz and Stegun, p. 230, 5.1.20</ref> :<math>\operatorname{Li}(x)-\pi(x) = O(\sqrt{x}\log x)</math> == See also == * [[Jørgen Pedersen Gram]] * [[Skewes' number]] == References == <references/> {{AS ref\|5\|228}} {{dlmf\|id=6\|title=Exponential, Logarithmic, Sine, and Cosine Integrals\|first=N. M. \|last=Temme}} {{Nonelementary Integral}} [[Category:Special hypergeometric functions]] [[Category:Integrals]]
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LaTeX to CAS translator

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SymPy

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Maple

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