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 L(s,\chi)=\prod_p\left(1-\chi(p)p^{-s}\right)^{-1}\text{ for }\text{Re}(s) > 1,}

${\displaystyle L(s,\chi )=\prod _{p}\left(1-\chi (p)p^{-s}\right)^{-1}{\text{ for }}{\text{Re}}(s)>1,}$

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

Semantic latex: \DirichletL@{s}{\Dirichletchar@{n}{k}} = \prod_p(1 - \Dirichletchar@@{p}{k} p^{-s})^{-1} for{Re}(s) > 1

Confidence: 0.59785840558897

Mathematica

Translation:

Information

Symbol info

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

Tests

Symbolic

Numeric

SymPy

Translation:

Information

Symbol info

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

Tests

Symbolic

Numeric

Maple

Translation:

Information

Symbol info

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

Tests

Symbolic

Numeric

Dependency Graph Information

Includes

$\chi$

$L$

$L(s,\chi )$

$s$

$L(s,\chi )$

$\zeta (s,q)$

Description

prime number

Euler product in the half-plane

function

product

Dirichlet character

absolute convergence

Complete translation information:

{
  "id" : "FORMULA_aaf3494b07ce4e989efe1114b80bfce1",
  "formula" : "L(s,\\chi)=\\prod_p\\left(1-\\chi(p)p^{-s}\\right)^{-1}\\text{ for }\\text{Re}(s) > 1",
  "semanticFormula" : "\\DirichletL@{s}{\\Dirichletchar@{n}{k}} = \\prod_p(1 - \\Dirichletchar@@{p}{k} p^{-s})^{-1} for{Re}(s) > 1",
  "confidence" : 0.5978584055889739,
  "translations" : {
    "Mathematica" : {
      "translation" : "",
      "translationInformation" : {
        "tokenTranslations" : {
          "Error" : "(LaTeX -> Mathematica) No translation possible for given token: Cannot extract information from feature set: \\DirichletL [\\DirichletL]"
        }
      }
    },
    "SymPy" : {
      "translation" : "",
      "translationInformation" : {
        "tokenTranslations" : {
          "Error" : "(LaTeX -> SymPy) No translation possible for given token: Cannot extract information from feature set: \\DirichletL [\\DirichletL]"
        }
      }
    },
    "Maple" : {
      "translation" : "",
      "translationInformation" : {
        "tokenTranslations" : {
          "Error" : "(LaTeX -> Maple) No translation possible for given token: Cannot extract information from feature set: \\DirichletL [\\DirichletL]"
        }
      }
    }
  },
  "positions" : [ {
    "section" : 2,
    "sentence" : 0,
    "word" : 28
  } ],
  "includes" : [ "\\chi", "L", "L(s, \\chi)", "s", "L(s,\\chi)", "\\zeta(s,q)" ],
  "isPartOf" : [ ],
  "definiens" : [ {
    "definition" : "prime number",
    "score" : 0.7125985104912714
  }, {
    "definition" : "Euler product in the half-plane",
    "score" : 0.6859086196238077
  }, {
    "definition" : "function",
    "score" : 0.6859086196238077
  }, {
    "definition" : "product",
    "score" : 0.6859086196238077
  }, {
    "definition" : "Dirichlet character",
    "score" : 0.6460746792928004
  }, {
    "definition" : "absolute convergence",
    "score" : 0.5988174995334326
  } ]
}

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

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Symbolic

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SymPy

Information

Tests

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Maple

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Numeric

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