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 \mathrm{kei}(x) = -\ln\left(\frac{x}{2}\right) \mathrm{bei}(x) - \frac{\pi}{4}\mathrm{ber}(x) + \sum_{k \geq 0} (-1)^k \frac{\psi(2k + 2)}{[(2k+1)!]^2} \left(\frac{x^2}{4}\right)^{2k+1}}

${\displaystyle \mathrm {kei} (x)=-\ln \left({\frac {x}{2}}\right)\mathrm {bei} (x)-{\frac {\pi }{4}}\mathrm {ber} (x)+\sum _{k\geq 0}(-1)^{k}{\frac {\psi (2k+2)}{[(2k+1)!]^{2}}}\left({\frac {x^{2}}{4}}\right)^{2k+1}}$

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

Semantic latex: \mathrm{kei}(x) = - \ln(\frac{x}{2}) \mathrm{bei}(x) - \frac{\cpi}{4} \mathrm{ber}(x) + \sum_{k \geq 0}(- 1)^k \frac{\digamma@{2 k + 2}}{[(2k+1)!]^2}(\frac{x^2}{4})^{2k+1}

Confidence: 0.6805

Mathematica

Translation: kei[x] == - Log[Divide[x,2]]*bei[x]-Divide[Pi,4]*ber[x]+ Sum[(- 1)^(k)*Divide[PolyGamma[2*k + 2],((2*k + 1)!)^(2)]*(Divide[(x)^(2),4])^(2*k + 1), {k, 0, Infinity}, GenerateConditions->None]

Information

Sub Equations

kei[x] = - Log[Divide[x,2]]*bei[x]-Divide[Pi,4]*ber[x]+ Sum[(- 1)^(k)*Divide[PolyGamma[2*k + 2],((2*k + 1)!)^(2)]*(Divide[(x)^(2),4])^(2*k + 1), {k, 0, Infinity}, GenerateConditions->None]

Free variables

x

Symbol info

Pi was translated to: Pi

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

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

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

Digamma / Psi function; Example: \digamma@{z}

Will be translated to: PolyGamma[$0] Constraints: z not element of {0, -1, -2, ...} Relevant links to definitions: DLMF: http://dlmf.nist.gov/5.2#E2 Mathematica: https://reference.wolfram.com/language/ref/PolyGamma.html

Natural logarithm; Example: \ln@@{z}

Will be translated to: Log[$0] Constraints: z != 0 Branch Cuts: (-\infty, 0] Relevant links to definitions: DLMF: http://dlmf.nist.gov/4.2#E2 Mathematica: https://reference.wolfram.com/language/ref/Log.html

Tests

Symbolic

Test expression: (kei[x])-(- Log[Divide[x,2]]*bei[x]-Divide[Pi,4]*ber[x]+ Sum[(- 1)^(k)*Divide[PolyGamma[2*k + 2],((2*k + 1)!)^(2)]*(Divide[(x)^(2),4])^(2*k + 1), {k, 0, Infinity}, GenerateConditions->None])

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: \digamma [\digamma]

Tests

Symbolic

Numeric

Maple

Translation: kei(x) = - ln((x)/(2))*bei(x)-(Pi)/(4)*ber(x)+ sum((- 1)^(k)*(Psi(2*k + 2))/((factorial(2*k + 1))^(2))*(((x)^(2))/(4))^(2*k + 1), k = 0..infinity)

Information

Sub Equations

kei(x) = - ln((x)/(2))*bei(x)-(Pi)/(4)*ber(x)+ sum((- 1)^(k)*(Psi(2*k + 2))/((factorial(2*k + 1))^(2))*(((x)^(2))/(4))^(2*k + 1), k = 0..infinity)

Free variables

x

Symbol info

Pi was translated to: Pi

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

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

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

Digamma / Psi function; Example: \digamma@{z}

Will be translated to: Psi($0) Constraints: z not element of {0, -1, -2, ...} Relevant links to definitions: DLMF: http://dlmf.nist.gov/5.2#E2 Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=Psi

Natural logarithm; Example: \ln@@{z}

Will be translated to: ln($0) Constraints: z != 0 Branch Cuts: (-\infty, 0] Relevant links to definitions: DLMF: http://dlmf.nist.gov/4.2#E2 Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=ln

Tests

Symbolic

Numeric

Dependency Graph Information

Includes

$\psi (z)$

$x)$

$x$

Description

series expansion

kei

asymptotic series

integer

special case kei

ker

Complete translation information:

{
  "id" : "FORMULA_4a08ce875ebda0e2c6e50935dbd7ba9e",
  "formula" : "\\mathrm{kei}(x) = -\\ln\\left(\\frac{x}{2}\\right) \\mathrm{bei}(x) - \\frac{\\pi}{4}\\mathrm{ber}(x) + \\sum_{k \\geq 0} (-1)^k \\frac{\\psi(2k + 2)}{[(2k+1)!]^2} \\left(\\frac{x^2}{4}\\right)^{2k+1}",
  "semanticFormula" : "\\mathrm{kei}(x) = - \\ln(\\frac{x}{2}) \\mathrm{bei}(x) - \\frac{\\cpi}{4} \\mathrm{ber}(x) + \\sum_{k \\geq 0}(- 1)^k \\frac{\\digamma@{2 k + 2}}{[(2k+1)!]^2}(\\frac{x^2}{4})^{2k+1}",
  "confidence" : 0.6805,
  "translations" : {
    "Mathematica" : {
      "translation" : "kei[x] == - Log[Divide[x,2]]*bei[x]-Divide[Pi,4]*ber[x]+ Sum[(- 1)^(k)*Divide[PolyGamma[2*k + 2],((2*k + 1)!)^(2)]*(Divide[(x)^(2),4])^(2*k + 1), {k, 0, Infinity}, GenerateConditions->None]",
      "translationInformation" : {
        "subEquations" : [ "kei[x] = - Log[Divide[x,2]]*bei[x]-Divide[Pi,4]*ber[x]+ Sum[(- 1)^(k)*Divide[PolyGamma[2*k + 2],((2*k + 1)!)^(2)]*(Divide[(x)^(2),4])^(2*k + 1), {k, 0, Infinity}, GenerateConditions->None]" ],
        "freeVariables" : [ "x" ],
        "tokenTranslations" : {
          "\\cpi" : "Pi was translated to: Pi",
          "kei" : "Function without DLMF-Definition. We keep it like it is (but delete prefix \\ if necessary).",
          "ber" : "Function without DLMF-Definition. We keep it like it is (but delete prefix \\ if necessary).",
          "bei" : "Function without DLMF-Definition. We keep it like it is (but delete prefix \\ if necessary).",
          "\\digamma" : "Digamma / Psi function; Example: \\digamma@{z}\nWill be translated to: PolyGamma[$0]\nConstraints: z not element of {0, -1, -2, ...}\nRelevant links to definitions:\nDLMF:         http://dlmf.nist.gov/5.2#E2\nMathematica:  https://reference.wolfram.com/language/ref/PolyGamma.html",
          "\\ln" : "Natural logarithm; Example: \\ln@@{z}\nWill be translated to: Log[$0]\nConstraints: z != 0\nBranch Cuts: (-\\infty, 0]\nRelevant links to definitions:\nDLMF:         http://dlmf.nist.gov/4.2#E2\nMathematica:  https://reference.wolfram.com/language/ref/Log.html"
        }
      },
      "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" : "kei[x]",
          "rhs" : "- Log[Divide[x,2]]*bei[x]-Divide[Pi,4]*ber[x]+ Sum[(- 1)^(k)*Divide[PolyGamma[2*k + 2],((2*k + 1)!)^(2)]*(Divide[(x)^(2),4])^(2*k + 1), {k, 0, Infinity}, GenerateConditions->None]",
          "testExpression" : "(kei[x])-(- Log[Divide[x,2]]*bei[x]-Divide[Pi,4]*ber[x]+ Sum[(- 1)^(k)*Divide[PolyGamma[2*k + 2],((2*k + 1)!)^(2)]*(Divide[(x)^(2),4])^(2*k + 1), {k, 0, Infinity}, GenerateConditions->None])",
          "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: \\digamma [\\digamma]"
        }
      }
    },
    "Maple" : {
      "translation" : "kei(x) = - ln((x)/(2))*bei(x)-(Pi)/(4)*ber(x)+ sum((- 1)^(k)*(Psi(2*k + 2))/((factorial(2*k + 1))^(2))*(((x)^(2))/(4))^(2*k + 1), k = 0..infinity)",
      "translationInformation" : {
        "subEquations" : [ "kei(x) = - ln((x)/(2))*bei(x)-(Pi)/(4)*ber(x)+ sum((- 1)^(k)*(Psi(2*k + 2))/((factorial(2*k + 1))^(2))*(((x)^(2))/(4))^(2*k + 1), k = 0..infinity)" ],
        "freeVariables" : [ "x" ],
        "tokenTranslations" : {
          "\\cpi" : "Pi was translated to: Pi",
          "kei" : "Function without DLMF-Definition. We keep it like it is (but delete prefix \\ if necessary).",
          "ber" : "Function without DLMF-Definition. We keep it like it is (but delete prefix \\ if necessary).",
          "bei" : "Function without DLMF-Definition. We keep it like it is (but delete prefix \\ if necessary).",
          "\\digamma" : "Digamma / Psi function; Example: \\digamma@{z}\nWill be translated to: Psi($0)\nConstraints: z not element of {0, -1, -2, ...}\nRelevant links to definitions:\nDLMF:  http://dlmf.nist.gov/5.2#E2\nMaple: https://www.maplesoft.com/support/help/maple/view.aspx?path=Psi",
          "\\ln" : "Natural logarithm; Example: \\ln@@{z}\nWill be translated to: ln($0)\nConstraints: z != 0\nBranch Cuts: (-\\infty, 0]\nRelevant links to definitions:\nDLMF:  http://dlmf.nist.gov/4.2#E2\nMaple: https://www.maplesoft.com/support/help/maple/view.aspx?path=ln"
        }
      }
    }
  },
  "positions" : [ {
    "section" : 4,
    "sentence" : 0,
    "word" : 29
  } ],
  "includes" : [ "\\psi(z)", "x)", "x" ],
  "isPartOf" : [ ],
  "definiens" : [ {
    "definition" : "series expansion",
    "score" : 0.8869384888466118
  }, {
    "definition" : "kei",
    "score" : 0.7936682821800771
  }, {
    "definition" : "asymptotic series",
    "score" : 0.6954080343007951
  }, {
    "definition" : "integer",
    "score" : 0.6288842031023242
  }, {
    "definition" : "special case kei",
    "score" : 0.6288842031023242
  }, {
    "definition" : "ker",
    "score" : 0.5816270233429564
  } ]
}

Specify your own input

Formula input
Wikitext context	In applied mathematics, the '''Kelvin functions''' ber<sub>''ν''</sub>(''x'') and bei<sub>''ν''</sub>(''x'') are the [[real part\|real]] and [[imaginary part]]s, respectively, of :<math>J_\nu \left (x e^{\frac{3 \pi i}{4}} \right ),\,</math><!-- Do not delete "\," it improves display of formula on certain browsers. ---> where ''x'' is real, and {{math\|''J<sub>ν</sub>''(''z'')}}, is the ''ν''<sup>th</sup> order [[Bessel function]] of the first kind. Similarly, the functions ker<sub>ν</sub>(''x'') and kei<sub>ν</sub>(''x'') are the real and imaginary parts, respectively, of :<math>K_\nu \left (x e^{\frac{\pi i}{4}} \right ),\,</math> <!-- Do not delete "\," it improves display of formula on certain browsers. ---> where {{math\|''K<sub>ν</sub>''(''z'')}} is the ''ν''<sup>th</sup> order [[Bessel function#Modified Bessel functions\|modified Bessel function]] of the second kind. These functions are named after [[William Thomson, 1st Baron Kelvin]]. While the Kelvin functions are defined as the real and imaginary parts of Bessel functions with ''x'' taken to be real, the functions can be analytically continued for complex arguments {{math\|''xe''<sup>''iφ''</sup>, 0 ≤ ''φ'' < 2''π''.}} With the exception of ber<sub>''n''</sub>(''x'') and bei<sub>''n''</sub>(''x'') for integral ''n'', the Kelvin functions have a [[branch point]] at ''x'' = 0. Below, {{math\|Γ(''z'')}} is the [[gamma function]] and {{math\|''ψ''(''z'')}} is the [[digamma function]]. == ber(''x'') == [[Image:Mplwp Kelvin-ber.svg\|thumb\|right\|ber(''x'') for ''x'' between 0 and 20.]] [[Image:Mplwp Kelvin-ber-norm.svg\|thumb\|right\|<math>\mathrm{ber}(x) / e^{x/\sqrt{2}}</math> for ''x'' between 0 and 50.]] For integers ''n'', ber<sub>''n''</sub>(''x'') has the series expansion :<math>\mathrm{ber}_n(x) = \left(\frac{x}{2}\right)^n \sum_{k \geq 0} \frac{\cos\left[\left(\frac{3n}{4} + \frac{k}{2}\right)\pi\right]}{k! \Gamma(n + k + 1)} \left(\frac{x^2}{4}\right)^k ,</math> where {{math\|Γ(''z'')}} is the [[gamma function]]. The special case ber<sub>0</sub>(''x''), commonly denoted as just ber(''x''), has the series expansion :<math>\mathrm{ber}(x) = 1 + \sum_{k \geq 1} \frac{(-1)^k}{[(2k)!]^2} \left(\frac{x}{2} \right )^{4k}</math> and [[asymptotic series]] :<math>\mathrm{ber}(x) \sim \frac{e^{\frac{x}{\sqrt{2}}}}{\sqrt{2 \pi x}} \left (f_1(x) \cos \alpha + g_1(x) \sin \alpha \right ) - \frac{\mathrm{kei}(x)}{\pi}</math>, where :<math>\alpha = \frac{x}{\sqrt{2}} - \frac{\pi}{8},</math> :<math>f_1(x) = 1 + \sum_{k \geq 1} \frac{\cos(k \pi / 4)}{k! (8x)^k} \prod_{l = 1}^k (2l - 1)^2</math> :<math>g_1(x) = \sum_{k \geq 1} \frac{\sin(k \pi / 4)}{k! (8x)^k} \prod_{l = 1}^k (2l - 1)^2 .</math> == bei(''x'') == [[Image:Mplwp Kelvin-bei.svg\|thumb\|right\|bei(''x'') for ''x'' between 0 and 20.]] [[Image:Mplwp Kelvin-bei-norm.svg\|thumb\|right\|<math>\mathrm{bei}(x) / e^{x/\sqrt{2}}</math> for ''x'' between 0 and 50.]] For integers ''n'', bei<sub>''n''</sub>(''x'') has the series expansion :<math>\mathrm{bei}_n(x) = \left(\frac{x}{2}\right)^n \sum_{k \geq 0} \frac{\sin\left[\left(\frac{3n}{4} + \frac{k}{2}\right)\pi\right]}{k! \Gamma(n + k + 1)} \left(\frac{x^2}{4}\right)^k .</math> The special case bei<sub>0</sub>(''x''), commonly denoted as just bei(''x''), has the series expansion :<math>\mathrm{bei}(x) = \sum_{k \geq 0} \frac{(-1)^k }{[(2k+1)!]^2} \left(\frac{x}{2} \right )^{4k+2}</math> and asymptotic series :<math>\mathrm{bei}(x) \sim \frac{e^{\frac{x}{\sqrt{2}}}}{\sqrt{2 \pi x}} [f_1(x) \sin \alpha - g_1(x) \cos \alpha] - \frac{\mathrm{ker}(x)}{\pi},</math> where α, <math>f_1(x)</math>, and <math>g_1(x)</math> are defined as for ber(''x''). {{clear}} == ker(''x'') == [[Image:Mplwp Kelvin-ker.svg\|thumb\|right\|ker(''x'') for ''x'' between 0 and 14.]] [[Image:Mplwp Kelvin-ker-norm.svg\|thumb\|right\|<math>\mathrm{ker}(x) e^{x/\sqrt{2}}</math> for ''x'' between 0 and 50.]] For integers ''n'', ker<sub>''n''</sub>(''x'') has the (complicated) series expansion :<math>\begin{align} &\mathrm{ker}_n(x) = - \ln\left(\frac{x}{2}\right) \mathrm{ber}_n(x) + \frac{\pi}{4}\mathrm{bei}_n(x) \\ &+ \frac{1}{2} \left(\frac{x}{2}\right)^{-n} \sum_{k=0}^{n-1} \cos\left[\left(\frac{3n}{4} + \frac{k}{2}\right)\pi\right] \frac{(n-k-1)!}{k!} \left(\frac{x^2}{4}\right)^k \\ &+ \frac{1}{2} \left(\frac{x}{2}\right)^n \sum_{k \geq 0} \cos\left[\left(\frac{3n}{4} + \frac{k}{2}\right)\pi\right] \frac{\psi(k+1) + \psi(n + k + 1)}{k! (n+k)!} \left(\frac{x^2}{4}\right)^k . \end{align}</math> The special case ker<sub>0</sub>(''x''), commonly denoted as just ker(''x''), has the series expansion :<math>\mathrm{ker}(x) = -\ln\left(\frac{x}{2}\right) \mathrm{ber}(x) + \frac{\pi}{4}\mathrm{bei}(x) + \sum_{k \geq 0} (-1)^k \frac{\psi(2k + 1)}{[(2k)!]^2} \left(\frac{x^2}{4}\right)^{2k}</math> and the asymptotic series :<math>\mathrm{ker}(x) \sim \sqrt{\frac{\pi}{2x}} e^{-\frac{x}{\sqrt{2}}} [f_2(x) \cos \beta + g_2(x) \sin \beta],</math> where :<math>\beta = \frac{x}{\sqrt{2}} + \frac{\pi}{8},</math> :<math>f_2(x) = 1 + \sum_{k \geq 1} (-1)^k \frac{\cos(k \pi / 4)}{k! (8x)^k} \prod_{l = 1}^k (2l - 1)^2</math> :<math>g_2(x) = \sum_{k \geq 1} (-1)^k \frac{\sin(k \pi / 4)}{k! (8x)^k} \prod_{l = 1}^k (2l - 1)^2.</math> {{clear}} == kei(''x'') == [[Image:Mplwp Kelvin-kei.svg\|thumb\|right\|kei(''x'') for ''x'' between 0 and 14.]] [[Image:Mplwp Kelvin-kei-norm.svg\|thumb\|right\|<math>\mathrm{kei}(x) e^{x/\sqrt{2}}</math> for ''x'' between 0 and 50.]] For integer ''n'', kei<sub>''n''</sub>(''x'') has the series expansion :<math>\begin{align} &\mathrm{kei}_n(x) = - \ln\left(\frac{x}{2}\right) \mathrm{bei}_n(x) - \frac{\pi}{4}\mathrm{ber}_n(x) \\ &-\frac{1}{2} \left(\frac{x}{2}\right)^{-n} \sum_{k=0}^{n-1} \sin\left[\left(\frac{3n}{4} + \frac{k}{2}\right)\pi\right] \frac{(n-k-1)!}{k!} \left(\frac{x^2}{4}\right)^k \\ &+ \frac{1}{2} \left(\frac{x}{2}\right)^n \sum_{k \geq 0} \sin\left[\left(\frac{3n}{4} + \frac{k}{2}\right)\pi\right] \frac{\psi(k+1) + \psi(n + k + 1)}{k! (n+k)!} \left(\frac{x^2}{4}\right)^k . \end{align}</math> The special case kei<sub>0</sub>(''x''), commonly denoted as just kei(''x''), has the series expansion :<math>\mathrm{kei}(x) = -\ln\left(\frac{x}{2}\right) \mathrm{bei}(x) - \frac{\pi}{4}\mathrm{ber}(x) + \sum_{k \geq 0} (-1)^k \frac{\psi(2k + 2)}{[(2k+1)!]^2} \left(\frac{x^2}{4}\right)^{2k+1}</math> and the asymptotic series :<math>\mathrm{kei}(x) \sim -\sqrt{\frac{\pi}{2x}} e^{-\frac{x}{\sqrt{2}}} [f_2(x) \sin \beta + g_2(x) \cos \beta],</math> where ''β'', ''f''<sub>2</sub>(''x''), and ''g''<sub>2</sub>(''x'') are defined as for ker(''x''). {{clear}} == See also == * [[Bessel function]] == References == * {{Abramowitz_Stegun_ref\|9\|379}} {{dlmf\|first=F. W. J. \|last=Olver\|first2=L. C. \|last2=Maximon\|id=10\|title=Bessel functions}} == External links == Weisstein, Eric W. "Kelvin Functions." From MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/KelvinFunctions.html] * GPL-licensed C/C++ source code for calculating Kelvin functions at codecogs.com: [https://web.archive.org/web/20070407195618/http://www.codecogs.com/d-ox/maths/special/bessel/kelvin.php] [[Category:Special hypergeometric functions]] [[Category:William Thomson, 1st Baron Kelvin\|Functions]]
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LaTeX to CAS translator

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