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 z\mathbf{J}_{\nu-1}(z)+z\mathbf{J}_{\nu+1}(z)=2\nu\mathbf{J}_\nu(z)-\frac{2\sin\pi\nu}{\pi}}

${\displaystyle z\mathbf {J} _{\nu -1}(z)+z\mathbf {J} _{\nu +1}(z)=2\nu \mathbf {J} _{\nu }(z)-{\frac {2\sin \pi \nu }{\pi }}}$

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

Semantic latex: z \AngerJ{\nu-1}@{z} + z \AngerJ{\nu+1}@{z} = 2 \nu \AngerJ{\nu}@{z} - \frac{2 \sin \cpi \nu}{\cpi}

Confidence: 0.88957271447778

Mathematica

Translation: z*AngerJ[\[Nu]- 1, z]+ z*AngerJ[\[Nu]+ 1, z] == 2*\[Nu]*AngerJ[\[Nu], z]-Divide[2*Sin[Pi]*\[Nu],Pi]

Information

Sub Equations

z*AngerJ[\[Nu]- 1, z]+ z*AngerJ[\[Nu]+ 1, z] = 2*\[Nu]*AngerJ[\[Nu], z]-Divide[2*Sin[Pi]*\[Nu],Pi]

Free variables

\[Nu]

z

Symbol info

Anger function; Example: \AngerJ{v}@{z}

Will be translated to: AngerJ[$0, $1] Relevant links to definitions: DLMF: http://dlmf.nist.gov/11.10#E1 Mathematica: https://reference.wolfram.com/language/ref/AngerJ.html

Pi was translated to: Pi

Sine; Example: \sin@@{z}

Will be translated to: Sin[$0] Relevant links to definitions: DLMF: http://dlmf.nist.gov/4.14#E1 Mathematica: https://reference.wolfram.com/language/ref/Sin.html

Tests

Symbolic

Test expression: (z*AngerJ[\[Nu]- 1, z]+ z*AngerJ[\[Nu]+ 1, z])-(2*\[Nu]*AngerJ[\[Nu], z]-Divide[2*Sin[Pi]*\[Nu],Pi])

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: \AngerJ [\AngerJ]

Tests

Symbolic

Numeric

Maple

Translation: z*AngerJ(nu - 1, z)+ z*AngerJ(nu + 1, z) = 2*nu*AngerJ(nu, z)-(2*sin(Pi)*nu)/(Pi)

Information

Sub Equations

z*AngerJ(nu - 1, z)+ z*AngerJ(nu + 1, z) = 2*nu*AngerJ(nu, z)-(2*sin(Pi)*nu)/(Pi)

Free variables

nu

z

Symbol info

Anger function; Example: \AngerJ{v}@{z}

Will be translated to: AngerJ($0, $1) Relevant links to definitions: DLMF: http://dlmf.nist.gov/11.10#E1 Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=AngerJ

Pi was translated to: Pi

Sine; Example: \sin@@{z}

Will be translated to: sin($0) Relevant links to definitions: DLMF: http://dlmf.nist.gov/4.14#E1 Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=sin

Tests

Symbolic

Numeric

Dependency Graph Information

Includes

$J_{\nu }$

$\mathbf {J} _{\nu }$

$\nu$

Description

inhomogeneous form of recurrence relation

Anger function

Weber function

Complete translation information:

{
  "id" : "FORMULA_d78016f4287d5d2c9cfb77ef07bc67f9",
  "formula" : "z\\mathbf{J}_{\\nu-1}(z)+z\\mathbf{J}_{\\nu+1}(z)=2\\nu\\mathbf{J}_\\nu(z)-\\frac{2\\sin\\pi\\nu}{\\pi}",
  "semanticFormula" : "z \\AngerJ{\\nu-1}@{z} + z \\AngerJ{\\nu+1}@{z} = 2 \\nu \\AngerJ{\\nu}@{z} - \\frac{2 \\sin \\cpi \\nu}{\\cpi}",
  "confidence" : 0.8895727144777772,
  "translations" : {
    "Mathematica" : {
      "translation" : "z*AngerJ[\\[Nu]- 1, z]+ z*AngerJ[\\[Nu]+ 1, z] == 2*\\[Nu]*AngerJ[\\[Nu], z]-Divide[2*Sin[Pi]*\\[Nu],Pi]",
      "translationInformation" : {
        "subEquations" : [ "z*AngerJ[\\[Nu]- 1, z]+ z*AngerJ[\\[Nu]+ 1, z] = 2*\\[Nu]*AngerJ[\\[Nu], z]-Divide[2*Sin[Pi]*\\[Nu],Pi]" ],
        "freeVariables" : [ "\\[Nu]", "z" ],
        "tokenTranslations" : {
          "\\AngerJ" : "Anger function; Example: \\AngerJ{v}@{z}\nWill be translated to: AngerJ[$0, $1]\nRelevant links to definitions:\nDLMF:         http://dlmf.nist.gov/11.10#E1\nMathematica:  https://reference.wolfram.com/language/ref/AngerJ.html",
          "\\cpi" : "Pi was translated to: Pi",
          "\\sin" : "Sine; Example: \\sin@@{z}\nWill be translated to: Sin[$0]\nRelevant links to definitions:\nDLMF:         http://dlmf.nist.gov/4.14#E1\nMathematica:  https://reference.wolfram.com/language/ref/Sin.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" : "z*AngerJ[\\[Nu]- 1, z]+ z*AngerJ[\\[Nu]+ 1, z]",
          "rhs" : "2*\\[Nu]*AngerJ[\\[Nu], z]-Divide[2*Sin[Pi]*\\[Nu],Pi]",
          "testExpression" : "(z*AngerJ[\\[Nu]- 1, z]+ z*AngerJ[\\[Nu]+ 1, z])-(2*\\[Nu]*AngerJ[\\[Nu], z]-Divide[2*Sin[Pi]*\\[Nu],Pi])",
          "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: \\AngerJ [\\AngerJ]"
        }
      }
    },
    "Maple" : {
      "translation" : "z*AngerJ(nu - 1, z)+ z*AngerJ(nu + 1, z) = 2*nu*AngerJ(nu, z)-(2*sin(Pi)*nu)/(Pi)",
      "translationInformation" : {
        "subEquations" : [ "z*AngerJ(nu - 1, z)+ z*AngerJ(nu + 1, z) = 2*nu*AngerJ(nu, z)-(2*sin(Pi)*nu)/(Pi)" ],
        "freeVariables" : [ "nu", "z" ],
        "tokenTranslations" : {
          "\\AngerJ" : "Anger function; Example: \\AngerJ{v}@{z}\nWill be translated to: AngerJ($0, $1)\nRelevant links to definitions:\nDLMF:  http://dlmf.nist.gov/11.10#E1\nMaple: https://www.maplesoft.com/support/help/maple/view.aspx?path=AngerJ",
          "\\cpi" : "Pi was translated to: Pi",
          "\\sin" : "Sine; Example: \\sin@@{z}\nWill be translated to: sin($0)\nRelevant links to definitions:\nDLMF:  http://dlmf.nist.gov/4.14#E1\nMaple: https://www.maplesoft.com/support/help/maple/view.aspx?path=sin"
        }
      }
    }
  },
  "positions" : [ {
    "section" : 4,
    "sentence" : 0,
    "word" : 10
  } ],
  "includes" : [ "J_{\\nu}", "\\mathbf{J}_{\\nu}", "\\nu" ],
  "isPartOf" : [ ],
  "definiens" : [ {
    "definition" : "inhomogeneous form of recurrence relation",
    "score" : 0.8869384888466118
  }, {
    "definition" : "Anger function",
    "score" : 0.6687181434333315
  }, {
    "definition" : "Weber function",
    "score" : 0.6687181434333315
  } ]
}

Specify your own input

Formula input
Wikitext context	In mathematics, the '''Anger function''', introduced by {{harvs\|txt\|authorlink=Carl Theodor Anger\|first=C. T.\|last=Anger\|year=1855}}, is a function defined as : <math>\mathbf{J}_\nu(z)=\frac{1}{\pi} \int_0^\pi \cos (\nu\theta-z\sin\theta) \,d\theta</math> and is closely related to [[Bessel function]]s. The '''Weber function''' (also known as '''Lommel-Weber function'''), introduced by {{harvs\|txt\|authorlink=Heinrich Friedrich Weber\|first=H. F.\|last=Weber\|year=1879}}, is a closely related function defined by : <math>\mathbf{E}_\nu(z)=\frac{1}{\pi} \int_0^\pi \sin (\nu\theta-z\sin\theta) \,d\theta</math> and is closely related to [[Bessel function]]s of the second kind. ==Relation between Weber and Anger functions== The Anger and Weber functions are related by :<math> \begin{align} \sin(\pi \nu)\mathbf{J}_\nu(z) &= \cos(\pi\nu)\mathbf{E}_\nu(z)-\mathbf{E}_{-\nu}(z) \\ -\sin(\pi \nu)\mathbf{E}_\nu(z) &= \cos(\pi\nu)\mathbf{J}_\nu(z)-\mathbf{J}_{-\nu}(z) \end{align} </math> so in particular if ν is not an integer they can be expressed as linear combinations of each other. If ν is an integer then Anger functions '''J'''<sub>ν</sub> are the same as Bessel functions ''J''<sub>ν</sub>, and Weber functions can be expressed as finite linear combinations of [[Struve function]]s. ==Power series expansion== The Anger function has the power series expansion<ref name=DLMF>{{dlmf\|id=11.10\|title=Anger-Weber Functions\|first=R. B. \|last=Paris}}</ref> :<math>\mathbf{J}_\nu(z)=\cos\frac{\pi\nu}{2}\sum_{k=0}^\infty\frac{(-1)^kz^{2k}}{4^k\Gamma\left(k+\frac{\nu}{2}+1\right)\Gamma\left(k-\frac{\nu}{2}+1\right)}+\sin\frac{\pi\nu}{2}\sum_{k=0}^\infty\frac{(-1)^kz^{2k+1}}{2^{2k+1}\Gamma\left(k+\frac{\nu}{2}+\frac{3}{2}\right)\Gamma\left(k-\frac{\nu}{2}+\frac{3}{2}\right)}</math> While the Weber function has the power series expansion<ref name=DLMF/> :<math>\mathbf{E}_\nu(z)=\sin\frac{\pi\nu}{2}\sum_{k=0}^\infty\frac{(-1)^kz^{2k}}{4^k\Gamma\left(k+\frac{\nu}{2}+1\right)\Gamma\left(k-\frac{\nu}{2}+1\right)}-\cos\frac{\pi\nu}{2}\sum_{k=0}^\infty\frac{(-1)^kz^{2k+1}}{2^{2k+1}\Gamma\left(k+\frac{\nu}{2}+\frac{3}{2}\right)\Gamma\left(k-\frac{\nu}{2}+\frac{3}{2}\right)}</math> ==Differential equations== The Anger and Weber functions are solutions of inhomogeneous forms of Bessel's equation :<math>z^2y^{\prime\prime} + zy^\prime +(z^2-\nu^2)y = 0 .</math> More precisely, the Anger functions satisfy the equation<ref name=DLMF/> :<math>z^2y^{\prime\prime} + zy^\prime +(z^2-\nu^2)y = \frac{(z-\nu)\sin(\pi \nu)}{\pi} ,</math> and the Weber functions satisfy the equation<ref name=DLMF/> :<math>z^2y^{\prime\prime} + zy^\prime +(z^2-\nu^2)y = -\frac{z+\nu+(z-\nu)\cos(\pi \nu)}{\pi}.</math> ==Recurrence Relations== The Anger function satisfies this inhomogeneous form of [[recurrence relation]]<ref name=DLMF/> :<math>z\mathbf{J}_{\nu-1}(z)+z\mathbf{J}_{\nu+1}(z)=2\nu\mathbf{J}_\nu(z)-\frac{2\sin\pi\nu}{\pi}</math> While the Weber function satisfies this inhomogeneous form of [[recurrence relation]]<ref name=DLMF/> :<math>z\mathbf{E}_{\nu-1}(z)+z\mathbf{E}_{\nu+1}(z)=2\nu\mathbf{E}_\nu(z)-\frac{2(1-\cos\pi\nu)}{\pi}</math> ==Delay Differential Equations== The Anger and Weber functions satisfy these homogeneous forms of [[delay differential equation]]s<ref name=DLMF/> :<math>\mathbf{J}_{\nu-1}(z)-\mathbf{J}_{\nu+1}(z)=2\dfrac{\partial}{\partial z}\mathbf{J}_\nu(z)</math> :<math>\mathbf{E}_{\nu-1}(z)-\mathbf{E}_{\nu+1}(z)=2\dfrac{\partial}{\partial z}\mathbf{E}_\nu(z)</math> The Anger and Weber functions also satisfy these inhomogeneous forms of [[delay differential equation]]s<ref name=DLMF/> :<math>z\dfrac{\partial}{\partial z}\mathbf{J}_\nu(z)\pm\nu\mathbf{J}_\nu(z)=\pm z\mathbf{J}_{\nu\mp1}(z)\pm\frac{\sin\pi\nu}{\pi}</math> :<math>z\dfrac{\partial}{\partial z}\mathbf{E}_\nu(z)\pm\nu\mathbf{E}_\nu(z)=\pm z\mathbf{E}_{\nu\mp1}(z)\pm\frac{1-\cos\pi\nu}{\pi}</math> ==References== {{AS ref\|12\|498}} C.T. Anger, Neueste Schr. d. Naturf. d. Ges. i. Danzig, 5 (1855) pp. 1–29 {{springer\|id=A/a012490\|title=Anger function\|first=A.P.\|last= Prudnikov\|authorlink=Anatolii Platonovich Prudnikov}} {{springer\|id=W/w097320\|title=Weber function\|first=A.P.\|last= Prudnikov}} [[G.N. Watson]], "A treatise on the theory of Bessel functions", 1–2, Cambridge Univ. Press (1952) H.F. Weber, Zurich Vierteljahresschrift, 24 (1879) pp. 33–76 {{Reflist}} [[Category:Special functions]]
	Purge cached results.

LaTeX to CAS translator

Mathematica

Information

Tests

Symbolic

Numeric

SymPy

Information

Tests

Symbolic

Numeric

Maple

Information

Tests

Symbolic

Numeric

Dependency Graph Information

Specify your own input

Navigation menu

Search