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 \Lambda(1-s,\overline{\chi})=\frac{i^ak^{1/2}}{\tau(\chi)}\Lambda(s,\chi),}

${\displaystyle \Lambda (1-s,{\overline {\chi }})={\frac {i^{a}k^{1/2}}{\tau (\chi )}}\Lambda (s,\chi ),}$

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

Semantic latex: \Lambda(1 - s , \Dirichletchar@{n}{k}) = \frac{\iunit^a k^{1/2}}{\tau(\Dirichletchar@{n}{k})} \Lambda(s , \Dirichletchar@{n}{k})

Confidence: 0.52121378887832

Mathematica

Translation: \[CapitalLambda][1 - s , DirichletCharacter[1, k, n]] == Divide[(I)^(a)* (k)^(1/2),\[Tau][DirichletCharacter[1, k, n]]]*\[CapitalLambda][s , DirichletCharacter[1, k, n]]

Information

Sub Equations

\[CapitalLambda][1 - s , DirichletCharacter[1, k, n]] = Divide[(I)^(a)* (k)^(1/2),\[Tau][DirichletCharacter[1, k, n]]]*\[CapitalLambda][s , DirichletCharacter[1, k, n]]

Free variables

\[CapitalLambda]

\[Tau]

a

k

n

s

Symbol info

Function without DLMF-Definition. We keep it like it is (but delete prefix \ if necessary).

Imaginary unit was translated to: I

Dirichlet character; Example: \Dirichletchar@@{n}{k}

Will be translated to: DirichletCharacter[1, $1, $0] Relevant links to definitions: DLMF: http://dlmf.nist.gov/27.8 Mathematica: https://reference.wolfram.com/language/ref/DirichletCharacter.html

Function without DLMF-Definition. We keep it like it is (but delete prefix \ if necessary).

Tests

Symbolic

Test expression: (\[CapitalLambda]*(1 - s , DirichletCharacter[1, k, n]))-(Divide[(I)^(a)* (k)^(1/2),\[Tau]*(DirichletCharacter[1, k, n])]*\[CapitalLambda]*(s , DirichletCharacter[1, k, n]))

ERROR:

{
    "result": "ERROR",
    "testTitle": "Simple",
    "testExpression": null,
    "resultExpression": null,
    "wasAborted": false,
    "conditionallySuccessful": false
}

Numeric

SymPy

Translation:

Information

Symbol info

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

Tests

Symbolic

Numeric

Maple

Translation:

Information

Symbol info

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

Tests

Symbolic

Numeric

Dependency Graph Information

Includes

$a$

$\chi$

$\tau (\chi )$

$s$

$\zeta (s,q)$

$L(s,\chi )$

$L(s,\chi )$

$k$

Complete translation information:

{
  "id" : "FORMULA_891bece57edc5d3a8c3b8ec8f74cbbce",
  "formula" : "\\Lambda(1-s,\\overline{\\chi})=\\frac{i^a k^{1/2}}{\\tau(\\chi)}\\Lambda(s,\\chi)",
  "semanticFormula" : "\\Lambda(1 - s , \\Dirichletchar@{n}{k}) = \\frac{\\iunit^a k^{1/2}}{\\tau(\\Dirichletchar@{n}{k})} \\Lambda(s , \\Dirichletchar@{n}{k})",
  "confidence" : 0.5212137888783196,
  "translations" : {
    "Mathematica" : {
      "translation" : "\\[CapitalLambda][1 - s , DirichletCharacter[1, k, n]] == Divide[(I)^(a)* (k)^(1/2),\\[Tau][DirichletCharacter[1, k, n]]]*\\[CapitalLambda][s , DirichletCharacter[1, k, n]]",
      "translationInformation" : {
        "subEquations" : [ "\\[CapitalLambda][1 - s , DirichletCharacter[1, k, n]] = Divide[(I)^(a)* (k)^(1/2),\\[Tau][DirichletCharacter[1, k, n]]]*\\[CapitalLambda][s , DirichletCharacter[1, k, n]]" ],
        "freeVariables" : [ "\\[CapitalLambda]", "\\[Tau]", "a", "k", "n", "s" ],
        "tokenTranslations" : {
          "\\Lambda" : "Function without DLMF-Definition. We keep it like it is (but delete prefix \\ if necessary).",
          "\\iunit" : "Imaginary unit was translated to: I",
          "\\Dirichletchar" : "Dirichlet character; Example: \\Dirichletchar@@{n}{k}\nWill be translated to: DirichletCharacter[1, $1, $0]\nRelevant links to definitions:\nDLMF:         http://dlmf.nist.gov/27.8\nMathematica:  https://reference.wolfram.com/language/ref/DirichletCharacter.html",
          "\\tau" : "Function without DLMF-Definition. We keep it like it is (but delete prefix \\ if necessary)."
        }
      },
      "numericResults" : {
        "overallResult" : "SKIPPED",
        "numberOfTests" : 0,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 0,
        "wasAborted" : false,
        "crashed" : false,
        "testCalculationsGroups" : [ ]
      },
      "symbolicResults" : {
        "overallResult" : "ERROR",
        "numberOfTests" : 1,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 1,
        "crashed" : false,
        "testCalculationsGroup" : [ {
          "lhs" : "\\[CapitalLambda]*(1 - s , DirichletCharacter[1, k, n])",
          "rhs" : "Divide[(I)^(a)* (k)^(1/2),\\[Tau]*(DirichletCharacter[1, k, n])]*\\[CapitalLambda]*(s , DirichletCharacter[1, k, n])",
          "testExpression" : "(\\[CapitalLambda]*(1 - s , DirichletCharacter[1, k, n]))-(Divide[(I)^(a)* (k)^(1/2),\\[Tau]*(DirichletCharacter[1, k, n])]*\\[CapitalLambda]*(s , DirichletCharacter[1, k, n]))",
          "testCalculations" : [ {
            "result" : "ERROR",
            "testTitle" : "Simple",
            "testExpression" : null,
            "resultExpression" : null,
            "wasAborted" : false,
            "conditionallySuccessful" : false
          } ]
        } ]
      }
    },
    "SymPy" : {
      "translation" : "",
      "translationInformation" : {
        "tokenTranslations" : {
          "Error" : "(LaTeX -> SymPy) No translation possible for given token: Cannot extract information from feature set: \\Dirichletchar [\\Dirichletchar]"
        }
      }
    },
    "Maple" : {
      "translation" : "",
      "translationInformation" : {
        "tokenTranslations" : {
          "Error" : "(LaTeX -> Maple) No translation possible for given token: Cannot extract information from feature set: \\Dirichletchar [\\Dirichletchar]"
        }
      }
    }
  },
  "positions" : [ ],
  "includes" : [ "a", "\\chi", "\\tau(\\chi)", "s", "\\zeta(s,q)", "L(s, \\chi)", "L(s,\\chi)", "k" ],
  "isPartOf" : [ ],
  "definiens" : [ ]
}

Specify your own input

Formula input
Wikitext context	{{DISPLAYTITLE:Dirichlet ''L''-function}} In [[mathematics]], a '''Dirichlet ''L''-series''' is a function of the form :<math>L(s,\chi) = \sum_{n=1}^\infty \frac{\chi(n)}{n^s}.</math> Here χ is a [[Dirichlet character]] and ''s'' a [[complex variable]] with [[real part]] greater than 1. By [[analytic continuation]], this function can be extended to a [[meromorphic function]] on the whole [[complex plane]], and is then called a '''Dirichlet ''L''-function''' and also denoted ''L''(''s'', χ). These functions are named after [[Peter Gustav Lejeune Dirichlet]] who introduced them in {{harv\|Dirichlet\|1837}} to prove the [[Dirichlet's theorem on arithmetic progressions\|theorem on primes in arithmetic progressions]] that also bears his name. In the course of the proof, Dirichlet shows that {{Nowrap\|''L''(''s'', χ)}} is non-zero at ''s'' = 1. Moreover, if χ is principal, then the corresponding Dirichlet ''L''-function has a [[simple pole]] at ''s'' = 1. == Zeros of the Dirichlet L-functions == If χ is a primitive character with χ(−1) = 1, then the only zeros of ''L''(''s'',χ) with Re(''s'') < 0 are at the negative even integers. If χ is a primitive character with χ(−1) = −1, then the only zeros of ''L''(''s'',χ) with Re(''s'') < 0 are at the negative odd integers. Up to the possible existence of a [[Siegel zero]], zero-free regions including and beyond the line Re(''s'') = 1 similar to that of the Riemann zeta function are known to exist for all Dirichlet ''L''-functions: for example, for χ a non-real character of modulus ''q'', we have :<math> \beta < 1 - \frac{c}{ \log \big(q(2+\|\gamma\|)\big)} \ </math> for β + iγ a non-real zero.<ref>{{cite book \| last=Montgomery \| first=Hugh L. \| author-link=Hugh Montgomery (mathematician) \| title=Ten lectures on the interface between analytic number theory and harmonic analysis \| series=Regional Conference Series in Mathematics \| volume=84 \| location=Providence, RI \| publisher=[[American Mathematical Society]] \| year=1994 \| isbn=0-8218-0737-4 \| zbl=0814.11001 \| page=163 }}</ref> Just as the Riemann zeta function is conjectured to obey the [[Riemann hypothesis]], so the Dirichlet ''L''-functions are conjectured to obey the [[generalized Riemann hypothesis]]. == Euler product == Since a Dirichlet character χ is [[completely multiplicative]], its ''L''-function can also be written as an [[Euler product]] in the [[half-plane]] of [[absolute convergence]]: :<math>L(s,\chi)=\prod_p\left(1-\chi(p)p^{-s}\right)^{-1}\text{ for }\text{Re}(s) > 1,</math> where the product is over all [[prime number]]s.<ref>{{harvnb\|Apostol\|1976\|loc=Theorem 11.7}}</ref> == Functional equation == Let us assume that χ is a primitive character to the modulus ''k''. Defining :<math>\Lambda(s,\chi) = \left(\frac{\pi}{k}\right)^{-(s+a)/2} \Gamma\left(\frac{s+a}{2}\right) L(s,\chi),</math> where Γ denotes the [[Gamma function]] and the symbol ''a'' is given by :<math>a=\begin{cases}0;&\mbox{if }\chi(-1)=1, \\ 1;&\mbox{if }\chi(-1)=-1,\end{cases}</math> one has the [[functional equation (L-function)\|functional equation]] :<math>\Lambda(1-s,\overline{\chi})=\frac{i^ak^{1/2}}{\tau(\chi)}\Lambda(s,\chi),</math> where τ(χ) is the [[Gauss sum]] :<math>\sum_{n=1}^k\chi(n)\exp(2\pi in/k).</math> Note that \|τ(χ)\| = ''k''<sup>1/2</sup>. == Relation to the Hurwitz zeta-function == The Dirichlet ''L''-functions may be written as a linear combination of the [[Hurwitz zeta function\|Hurwitz zeta-function]] at rational values. Fixing an integer ''k'' ≥ 1, the Dirichlet ''L''-functions for characters modulo ''k'' are linear combinations, with constant coefficients, of the ζ(''s'',''q'') where ''q'' = ''m''/''k'' and ''m'' = 1, 2, ..., ''k''. This means that the Hurwitz zeta-function for rational ''q'' has analytic properties that are closely related to the Dirichlet ''L''-functions. Specifically, let χ be a character modulo ''k''. Then we can write its Dirichlet ''L''-function as :<math>L(s,\chi) = \sum_{n=1}^\infty \frac {\chi(n)}{n^s} = \frac {1}{k^s} \sum_{m=1}^k \chi(m)\; \zeta \left(s,\frac{m}{k}\right).</math> ==See also== [[Generalized Riemann hypothesis]] [[L-function]] [[Modularity theorem]] [[Artin conjecture (L-functions)\|Artin conjecture]] [[Special values of L-functions]] ==Notes== {{reflist}} == References == {{Apostol IANT}} {{dlmf\|id=25.15\|first=T. M.\|last=Apostol}} {{cite book\|author=H. Davenport \| title=Multiplicative Number Theory \|publisher=Springer \|year=2000 \|isbn=0-387-95097-4}} * {{Cite journal \| last=Dirichlet \| first=P. G. L. \| author-link=Peter Gustav Lejeune Dirichlet \| title=Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält \| journal=Abhand. Ak. Wiss. Berlin \| volume=48 \| year=1837 \| ref=harv \| postscript=. }} * {{springer\|title=Dirichlet-L-function\|id=p/d032890}} {{L-functions-footer}} [[Category:Zeta and L-functions]]
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LaTeX to CAS translator

Mathematica

Information

Tests

Symbolic

Numeric

SymPy

Information

Tests

Symbolic

Numeric

Maple

Information

Tests

Symbolic

Numeric

Dependency Graph Information

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