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 \frac{d^2y}{dt^2} + f(t) y = 0, }

${\displaystyle {\frac {d^{2}y}{dt^{2}}}+f(t)y=0,}$

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

Semantic latex: \deriv [2]{y}{t} + f(t) y = 0

Confidence: 0

Mathematica

Translation: D[y, {t, 2}]+ f[t]* y == 0

Information

Sub Equations

D[y, {t, 2}]+ f[t]* y = 0

Free variables

t

y

Symbol info

Derivative; Example: \deriv[n]{f}{x}

Will be translated to: D[$1, {$2, $0}] Relevant links to definitions: DLMF: http://dlmf.nist.gov/1.4#E4 Mathematica: https://reference.wolfram.com/language/ref/D.html

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

Tests

Symbolic

Test expression: (D[y, {t, 2}]+ f*(t)*y)-(0)

ERROR:

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

Numeric

SymPy

Translation: diff(y, t, 2)+ f(t)* y == 0

Information

Sub Equations

diff(y, t, 2)+ f(t)* y = 0

Free variables

t

y

Symbol info

Derivative; Example: \deriv[n]{f}{x}

Will be translated to: diff($1, $2, $0) Relevant links to definitions: DLMF: http://dlmf.nist.gov/1.4#E4 SymPy: https://docs.sympy.org/latest/tutorial/calculus.html#derivatives

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

Tests

Symbolic

Numeric

Maple

Translation: diff(y, [t$(2)])+ f(t)* y = 0

Information

Sub Equations

diff(y, [t$(2)])+ f(t)* y = 0

Free variables

t

y

Symbol info

Derivative; Example: \deriv[n]{f}{x}

Will be translated to: diff($1, [$2$($0)]) Relevant links to definitions: DLMF: http://dlmf.nist.gov/1.4#E4 Maple: https://www.maplesoft.com/support/help/Maple/view.aspx?path=diff

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

Tests

Symbolic

Numeric

Dependency Graph Information

Includes

$f(t)$

$t$

Description

second-order linear ordinary differential equation

periodic function by minimal period

mathematics

Hill differential equation

Hill equation

Complete translation information:

{
  "id" : "FORMULA_6ef227c696b6ffa1b06a30923428b22f",
  "formula" : "\\frac{d^2y}{dt^2} + f(t) y = 0",
  "semanticFormula" : "\\deriv [2]{y}{t} + f(t) y = 0",
  "confidence" : 0.0,
  "translations" : {
    "Mathematica" : {
      "translation" : "D[y, {t, 2}]+ f[t]* y == 0",
      "translationInformation" : {
        "subEquations" : [ "D[y, {t, 2}]+ f[t]* y = 0" ],
        "freeVariables" : [ "t", "y" ],
        "tokenTranslations" : {
          "\\deriv1" : "Derivative; Example: \\deriv[n]{f}{x}\nWill be translated to: D[$1, {$2, $0}]\nRelevant links to definitions:\nDLMF:         http://dlmf.nist.gov/1.4#E4\nMathematica:  https://reference.wolfram.com/language/ref/D.html",
          "f" : "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" : "D[y, {t, 2}]+ f*(t)*y",
          "rhs" : "0",
          "testExpression" : "(D[y, {t, 2}]+ f*(t)*y)-(0)",
          "testCalculations" : [ {
            "result" : "ERROR",
            "testTitle" : "Simple",
            "testExpression" : null,
            "resultExpression" : null,
            "wasAborted" : false,
            "conditionallySuccessful" : false
          } ]
        } ]
      }
    },
    "SymPy" : {
      "translation" : "diff(y, t, 2)+ f(t)* y == 0",
      "translationInformation" : {
        "subEquations" : [ "diff(y, t, 2)+ f(t)* y = 0" ],
        "freeVariables" : [ "t", "y" ],
        "tokenTranslations" : {
          "\\deriv1" : "Derivative; Example: \\deriv[n]{f}{x}\nWill be translated to: diff($1, $2, $0)\nRelevant links to definitions:\nDLMF:  http://dlmf.nist.gov/1.4#E4\nSymPy: https://docs.sympy.org/latest/tutorial/calculus.html#derivatives",
          "f" : "Function without DLMF-Definition. We keep it like it is (but delete prefix \\ if necessary)."
        }
      }
    },
    "Maple" : {
      "translation" : "diff(y, [t$(2)])+ f(t)* y = 0",
      "translationInformation" : {
        "subEquations" : [ "diff(y, [t$(2)])+ f(t)* y = 0" ],
        "freeVariables" : [ "t", "y" ],
        "tokenTranslations" : {
          "\\deriv1" : "Derivative; Example: \\deriv[n]{f}{x}\nWill be translated to: diff($1, [$2$($0)])\nRelevant links to definitions:\nDLMF:  http://dlmf.nist.gov/1.4#E4\nMaple: https://www.maplesoft.com/support/help/Maple/view.aspx?path=diff",
          "f" : "Function without DLMF-Definition. We keep it like it is (but delete prefix \\ if necessary)."
        }
      }
    }
  },
  "positions" : [ {
    "section" : 0,
    "sentence" : 0,
    "word" : 21
  } ],
  "includes" : [ "f(t)", "t" ],
  "isPartOf" : [ ],
  "definiens" : [ {
    "definition" : "second-order linear ordinary differential equation",
    "score" : 0.722
  }, {
    "definition" : "periodic function by minimal period",
    "score" : 0.7125985104912714
  }, {
    "definition" : "mathematics",
    "score" : 0.6859086196238077
  }, {
    "definition" : "Hill differential equation",
    "score" : 0.6460746792928004
  }, {
    "definition" : "Hill equation",
    "score" : 0.6460746792928004
  } ]
}

Specify your own input

Formula input
Wikitext context	{{distinguish\|Hill equation (biochemistry)}} In [[mathematics]], the '''Hill equation''' or '''Hill differential equation''' is the second-order linear [[ordinary differential equation]] :<math> \frac{d^2y}{dt^2} + f(t) y = 0, </math> where <math> f(t)</math> is a [[periodic function]] by minimal period <math> \pi </math>. By these we mean that for all <math>t </math> :<math>f(t+\pi)=f(t), </math> and if <math> p</math> is a number with <math>0 < p < \pi </math>, the equation <math> f(t+p) = f(t) </math> must fail for some <math> t </math>.<ref>{{cite book\|last=Magnus\|first=W.\|last2=Winkler\|first2=S.\|title=Hill's equation\|year=2013\|publisher=Courier\|isbn=9780486150291}}</ref> It is named after [[George William Hill]], who introduced it in 1886.<ref>{{cite journal\|doi=10.1007/BF02417081\|authorlink=George William Hill\|first=G.W.\|last=Hill\|title=On the Part of the Motion of Lunar Perigee Which is a Function of the Mean Motions of the Sun and Moon\|journal=Acta Math.\|volume=8\|issue=1\|pages=1–36\|year=1886\|url=https://zenodo.org/record/1691491/files/article.pdf}}</ref> Because <math> f(t) </math> has period <math>\pi </math>, the Hill equation can be rewritten using the [[Fourier series]] of <math> f(t)</math>: :<math>\frac{d^2y}{dt^2}+\left(\theta_0+2\sum_{n=1}^\infty \theta_n \cos(2nt)+\sum_{m=1}^\infty \phi_m \sin(2mt) \right ) y=0. </math> Important special cases of Hill's equation include the [[Mathieu_function#Mathieu_equation\|Mathieu equation]] (in which only the terms corresponding to ''n'' = 0, 1 are included) and the [[Meissner equation]]. Hill's equation is an important example in the understanding of periodic differential equations. Depending on the exact shape of <math> f(t) </math>, solutions may stay bounded for all time, or the amplitude of the oscillations in solutions may grow exponentially.<ref>{{cite book \| last = Teschl \| given = Gerald \|authorlink=Gerald Teschl \| title = Ordinary Differential Equations and Dynamical Systems \| publisher=[[American Mathematical Society]] \| place = [[Providence, Rhode Island\|Providence]] \| year = 2012 \| isbn= 978-0-8218-8328-0 \| url = https://www.mat.univie.ac.at/~gerald/ftp/book-ode/}}</ref> The precise form of the solutions to Hill's equation is described by [[Floquet theory]]. Solutions can also be written in terms of Hill determinants. Aside from its original application to lunar stability, the Hill equation appears in many settings including the modeling of a [[quadrupole mass spectrometer]], as the one-dimensional [[Schrödinger equation]] of an electron in a crystal, [[quantum optics]] of two-level systems, and in [[Accelerator_physics#Beam_dynamics\|accelerator physics]]. ==References== {{Reflist}} <!--- * [[Heinrich Guggenheimer]] (1977) ''Applicable Geometry'', pages 73–98, Krieger, Huntington ISBN 0-88275-368-1 . --> ==External links== * {{springer\|title=Hill equation\|id=p/h047440}} * {{mathworld\|urlname=HillsDifferentialEquation\|title=Hill's Differential Equation}} *{{dlmf\|first=G.\|last=Wolf\|id=28.29\|title=Mathieu Functions and Hill’s Equation}} [[Category:Ordinary differential equations]] {{mathapplied-stub}}
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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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