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 t }

${\displaystyle t}$

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

Semantic latex: t

Confidence: 0

Mathematica

Translation: t

Information

Sub Equations

t

Free variables

t

Tests

Symbolic

Numeric

SymPy

Translation: t

Information

Sub Equations

t

Free variables

t

Tests

Symbolic

Numeric

Maple

Translation: t

Information

Sub Equations

t

Free variables

t

Tests

Symbolic

Numeric

Dependency Graph Information

Is part of

$f(t)$

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

$f(t+\pi )=f(t)$

$f(t+p)=f(t)$

Description

Hill equation

equation

second-order linear ordinary differential equation

periodic function by minimal period

number

mathematics

Fourier series

Hill differential equation

period

exact shape

solution

amplitude of the oscillation

time

Complete translation information:

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  "includes" : [ ],
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  "definiens" : [ {
    "definition" : "Hill equation",
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    "definition" : "number",
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    "definition" : "Fourier series",
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  }, {
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  }, {
    "definition" : "period",
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    "definition" : "exact shape",
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  }, {
    "definition" : "solution",
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  }, {
    "definition" : "amplitude of the oscillation",
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  }, {
    "definition" : "time",
    "score" : 0.2879504737342633
  } ]
}

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

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SymPy

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Maple

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Tests

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Dependency Graph Information

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