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^2y^{\prime\prime} + zy^\prime +(z^2-\nu^2)y = 0 .}

${\displaystyle z^{2}y^{\prime \prime }+zy^{\prime }+(z^{2}-\nu ^{2})y=0.}$

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

Semantic latex: z^2y^{\prime\prime} + zy^\prime +(z^2-\nu^2)y = 0

Confidence: 0

Mathematica

Translation:

Information

Symbol info

(LaTeX -> Mathematica) The input LaTeX is invalid: Primes can only be translated behind semantic macros (differentiation primes) but not in other places.

Tests

Symbolic

Numeric

SymPy

Translation:

Information

Symbol info

(LaTeX -> SymPy) The input LaTeX is invalid: Primes can only be translated behind semantic macros (differentiation primes) but not in other places.

Tests

Symbolic

Numeric

Maple

Translation:

Information

Symbol info

(LaTeX -> Maple) The input LaTeX is invalid: Primes can only be translated behind semantic macros (differentiation primes) but not in other places.

Tests

Symbolic

Numeric

Dependency Graph Information

Includes

$\nu$

Description

Bessel 's equation

solution of inhomogeneous form

anger

Weber function

Complete translation information:

{
  "id" : "FORMULA_2f97e5e1cc6d5281d4d1c9cae1601201",
  "formula" : "z^2y^{\\prime\\prime} + zy^\\prime +(z^2-\\nu^2)y = 0",
  "semanticFormula" : "z^2y^{\\prime\\prime} + zy^\\prime +(z^2-\\nu^2)y = 0",
  "confidence" : 0.0,
  "translations" : {
    "Mathematica" : {
      "translation" : "",
      "translationInformation" : {
        "tokenTranslations" : {
          "Error" : "(LaTeX -> Mathematica) The input LaTeX is invalid: Primes can only be translated behind semantic macros (differentiation primes) but not in other places."
        }
      }
    },
    "SymPy" : {
      "translation" : "",
      "translationInformation" : {
        "tokenTranslations" : {
          "Error" : "(LaTeX -> SymPy) The input LaTeX is invalid: Primes can only be translated behind semantic macros (differentiation primes) but not in other places."
        }
      }
    },
    "Maple" : {
      "translation" : "",
      "translationInformation" : {
        "tokenTranslations" : {
          "Error" : "(LaTeX -> Maple) The input LaTeX is invalid: Primes can only be translated behind semantic macros (differentiation primes) but not in other places."
        }
      }
    }
  },
  "positions" : [ {
    "section" : 3,
    "sentence" : 0,
    "word" : 14
  } ],
  "includes" : [ "\\nu" ],
  "isPartOf" : [ ],
  "definiens" : [ {
    "definition" : "Bessel 's equation",
    "score" : 0.722
  }, {
    "definition" : "solution of inhomogeneous form",
    "score" : 0.6859086196238077
  }, {
    "definition" : "anger",
    "score" : 0.6460746792928004
  }, {
    "definition" : "Weber function",
    "score" : 0.5988174995334326
  } ]
}

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]]
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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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