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 P_n(x)}
... is translated to the CAS output ...
Semantic latex: \LegendrepolyP{n}@{x}
Confidence: 0.72556733422884
Mathematica
Translation: LegendreP[n, x]
Information
Sub Equations
- LegendreP[n, x]
Free variables
- n
- x
Symbol info
- Legendre polynomial; Example: \LegendrepolyP{n}@{x}
Will be translated to: LegendreP[$0, $1] Relevant links to definitions: DLMF: http://dlmf.nist.gov/18.3#T1.t1.r10 Mathematica: https://reference.wolfram.com/language/ref/LegendreP.html
Tests
Symbolic
Numeric
SymPy
Translation:
Information
Symbol info
- (LaTeX -> SymPy) No translation possible for given token: Cannot extract information from feature set: \LegendrepolyP [\LegendrepolyP]
Tests
Symbolic
Numeric
Maple
Translation: LegendreP(n, x)
Information
Sub Equations
- LegendreP(n, x)
Free variables
- n
- x
Symbol info
- Legendre polynomial; Example: \LegendrepolyP{n}@{x}
Will be translated to: LegendreP($0, $1) Relevant links to definitions: DLMF: http://dlmf.nist.gov/18.3#T1.t1.r10 Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=LegendreP
Tests
Symbolic
Numeric
Dependency Graph Information
Includes
Is part of
- Failed to parse (syntax error): {\displaystyle P\~_{n}(x)}
Description
- standardization
- polynomial of degree
- polynomial
- overall scale factor
- standardized polynomial of degree
- constructive definition
- Legendre polynomial
- fix
- orthogonality
- expansion
- form
- Bonnet 's recursion formula
- property
- zero
- function
- coefficient
- respect
- condition
- Chebyshev polynomial
- endpoint
- solution
- argument of magnitude
- axis of symmetry
- eigenfunction
- explicit representation
- expression
- fact
- first few Legendre polynomial
- Legendre 's differential equation
- norm over the interval
- normalization of the Legendre polynomial
- sequence of sum
- single equation
- solution for the potential
- term of solution
- useful property
- normalization
- power
- coefficient of power
- above one
- angle between the position
- Askey -- Gasper inequality for Legendre polynomial
- Bessel function
- eigenvalue
- graph of these polynomial
- integer
- Legendre polynomial by simple monomial
- local minima
- norm on the interval
- orthogonality relation
- parity
- polar angle
- potential
- rational Legendre function of degree
- Rodrigues ' formula
- third definition
- three-term recurrence relation
- trigonometric function
- value
- value at the boundary
- interval
- actual norm
- Adrien-Marie Legendre as the coefficient
- alternative expression
- axis
- compact expression for the Legendre polynomial
- completeness property
- first several order
- form of the binomial coefficient
- integration of Legendre polynomial
- Kronecker delta
- Legendre polynomial of a scalar product
- maximum
- mean
- multipole expansion
- one
- origin
- parity property
- piecewise continuous function
- radius
- series for this solution
- spherical harmonic
- statement
- zenith angle
- unit vector
- coefficient in a formal expansion
- affine transformation
- i.e.
- many discontinuity in the interval
- observation point
- observer
- recursion formula
- respect to the same norm
- spherical coordinate
- side
- angle
- article
- differential equation
- function of the form
- Legendre polynomials ' definition
- length of the vector
- scaling
- orthogonality property
- vector
Complete translation information:
{
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"formula" : "P_n(x)",
"semanticFormula" : "\\LegendrepolyP{n}@{x}",
"confidence" : 0.7255673342288351,
"translations" : {
"Mathematica" : {
"translation" : "LegendreP[n, x]",
"translationInformation" : {
"subEquations" : [ "LegendreP[n, x]" ],
"freeVariables" : [ "n", "x" ],
"tokenTranslations" : {
"\\LegendrepolyP" : "Legendre polynomial; Example: \\LegendrepolyP{n}@{x}\nWill be translated to: LegendreP[$0, $1]\nRelevant links to definitions:\nDLMF: http://dlmf.nist.gov/18.3#T1.t1.r10\nMathematica: https://reference.wolfram.com/language/ref/LegendreP.html"
}
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"SymPy" : {
"translation" : "",
"translationInformation" : {
"tokenTranslations" : {
"Error" : "(LaTeX -> SymPy) No translation possible for given token: Cannot extract information from feature set: \\LegendrepolyP [\\LegendrepolyP]"
}
},
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},
"Maple" : {
"translation" : "LegendreP(n, x)",
"translationInformation" : {
"subEquations" : [ "LegendreP(n, x)" ],
"freeVariables" : [ "n", "x" ],
"tokenTranslations" : {
"\\LegendrepolyP" : "Legendre polynomial; Example: \\LegendrepolyP{n}@{x}\nWill be translated to: LegendreP($0, $1)\nRelevant links to definitions:\nDLMF: http://dlmf.nist.gov/18.3#T1.t1.r10\nMaple: https://www.maplesoft.com/support/help/maple/view.aspx?path=LegendreP"
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"includes" : [ "n", "P_n", "P_m", "x", "P_{n}(x)", "Q_n", "P" ],
"isPartOf" : [ "\\int_{-1}^1 P_m(x) P_n(x)\\,dx = 0 \\quad \\text{if } n \\ne m", "P_n(1) = 1", "P_0(x) = 1", "P_1(x)", "P_1(x) = x", "P_2(x)", "\\frac{1}{\\sqrt{1-2xt+t^2}} = \\sum_{n=0}^\\infty P_n(x) t^n \\,", "P_0(x) = 1 \\,,\\quad P_1(x) = x", "\\frac{x-t}{\\sqrt{1-2xt+t^2}} = \\left(1-2xt+t^2\\right) \\sum_{n=1}^\\infty n P_n(x) t^{n-1} \\,", "(n+1) P_{n+1}(x) = (2n+1) x P_n(x) - n P_{n-1}(x)\\,", "\\frac{d}{dx}\\left[\\left(1-x^2\\right) \\frac{dP_n(x)}{dx}\\right] + n(n+1) P_n(x) = 0\\,", "P_{n}(x)", "P_n(\\cos\\theta)", "\\int_{-1}^1 P_m(x) P_n(x)\\,dx = \\frac{2}{2n + 1} \\delta_{mn}", "f_n(x) = \\sum_{\\ell=0}^n a_\\ell P_\\ell(x)", "a_\\ell = \\frac{2\\ell + 1}{2} \\int_{-1}^1 f(x) P_\\ell(x)\\,dx", "\\sum_{\\ell=0}^\\infty \\frac{2\\ell + 1}{2} P_\\ell(x)P_\\ell(y) = \\delta(x-y)", "P_n(x) = \\frac{1}{2^n n!} \\frac{d^n}{dx^n} (x^2 -1)^n \\,", "\\begin{align}P_n(x)&= \\frac{1}{2^n} \\sum_{k=0}^n \\binom{n}{k}^2 (x-1)^{n-k}(x+1)^k, \\\\P_n(x)&=\\sum_{k=0}^n \\binom{n}{k} \\binom{n+k}{k} \\left( \\frac{x-1}{2} \\right)^k, \\\\P_n(x)&=\\frac1{2^n}\\sum_{k=0}^{[\\frac n2]}(-1)^k\\binom nk\\binom{2n-2k}n x^{n-2k},\\\\P_n(x)&= 2^n \\sum_{k=0}^n x^k \\binom{n}{k} \\binom{\\frac{n+k-1}{2}}{n},\\end{align}", "\\begin{array}{r|r} n & P_n(x) \\\\\\hline 0 & 1 \\\\ 1 & x \\\\ 2 & \\tfrac12 \\left(3x^2-1\\right) \\\\ 3 & \\tfrac12 \\left(5x^3-3x\\right) \\\\ 4 & \\tfrac18 \\left(35x^4-30x^2+3\\right) \\\\ 5 & \\tfrac18 \\left(63x^5-70x^3+15x\\right) \\\\ 6 & \\tfrac1{16} \\left(231x^6-315x^4+105x^2-5\\right) \\\\ 7 & \\tfrac1{16} \\left(429x^7-693x^5+315x^3-35x\\right) \\\\ 8 & \\tfrac1{128} \\left(6435x^8-12012x^6+6930x^4-1260x^2+35\\right) \\\\ 9 & \\tfrac1{128} \\left(12155x^9-25740x^7+18018x^5-4620x^3+315x\\right) \\\\10 & \\tfrac1{256} \\left(46189x^{10}-109395x^8+90090x^6-30030x^4+3465x^2-63\\right) \\\\\\hline\\end{array}", "\\frac{1}{\\left| \\mathbf{x}-\\mathbf{x}' \\right|} = \\frac{1}{\\sqrt{r^2+{r'}^2-2r{r'}\\cos\\gamma}} = \\sum_{\\ell=0}^\\infty \\frac{{r'}^\\ell}{r^{\\ell+1}} P_\\ell(\\cos \\gamma)", "\\Phi(r,\\theta) = \\sum_{\\ell=0}^\\infty \\left( A_\\ell r^\\ell + B_\\ell r^{-(\\ell+1)} \\right) P_\\ell(\\cos\\theta) \\,", "\\frac{1}{\\sqrt{1 + \\eta^2 - 2\\eta x}} = \\sum_{k=0}^\\infty \\eta^k P_k(x)", "\\Phi(r, \\theta) \\propto \\frac{1}{r} \\sum_{k=0}^\\infty \\left( \\frac{a}{r} \\right)^k P_k(\\cos \\theta)", "T_{n}(cos \\theta) \\equiv cos n\\theta", "P_{n}(cos \\theta)", "\\begin{align}T_0(\\cos\\theta)&=1 &&=P_0(\\cos\\theta),\\\\[4pt]T_1(\\cos\\theta)&=\\cos \\theta&&=P_1(\\cos\\theta),\\\\[4pt]T_2(\\cos\\theta)&=\\cos 2\\theta&&=\\tfrac{1}{3}\\bigl(4P_2(\\cos\\theta)-P_0(\\cos\\theta)\\bigr),\\\\[4pt]T_3(\\cos\\theta)&=\\cos 3\\theta&&=\\tfrac{1}{5}\\bigl(8P_3(\\cos\\theta)-3P_1(\\cos\\theta)\\bigr),\\\\[4pt]T_4(\\cos\\theta)&=\\cos 4\\theta&&=\\tfrac{1}{105}\\bigl(192P_4(\\cos\\theta)-80P_2(\\cos\\theta)-7P_0(\\cos\\theta)\\bigr),\\\\[4pt]T_5(\\cos\\theta)&=\\cos 5\\theta&&=\\tfrac{1}{63}\\bigl(128P_5(\\cos\\theta)-56P_3(\\cos\\theta)-9P_1(\\cos\\theta)\\bigr),\\\\[4pt]T_6(\\cos\\theta)&=\\cos 6\\theta&&=\\tfrac{1}{1155}\\bigl(2560P_6(\\cos\\theta)-1152P_4(\\cos\\theta)-220P_2(\\cos\\theta)-33P_0(\\cos\\theta)\\bigr).\\end{align}", "\\frac{\\sin (n+1)\\theta}{\\sin\\theta}=\\sum_{\\ell=0}^n P_\\ell(\\cos\\theta) P_{n-\\ell}(\\cos\\theta)", "P_n(-x) = (-1)^n P_n(x) \\,", "\\int_{-1}^1 P_n(x)\\,dx = 0 \\text{ for } n\\ge1", "P_n(1) = 1 \\,", "\\sum_{j=0}^n P_j(x) \\ge 0 \\,\\quad \\text{for } x\\ge -1 \\,", "P_\\ell \\left(r \\cdot r'\\right) = \\frac{4\\pi}{2\\ell + 1} \\sum_{m=-\\ell}^\\ell Y_{\\ell m}(\\theta,\\varphi) Y_{\\ell m}^*(\\theta',\\varphi')\\,", "(n+1) P_{n+1}(x) = (2n+1) x P_n(x) - n P_{n-1}(x)", "\\frac{x^2-1}{n} \\frac{d}{dx} P_n(x) = xP_n(x) - P_{n-1}(x) ", "\\frac{d}{dx} P_{n+1}(x) = (n+1)P_n(x) + x \\frac{d}{dx}P_{n}(x) \\,", "(2n+1) P_n(x) = \\frac{d}{dx} \\bigl( P_{n+1}(x) - P_{n-1}(x) \\bigr) \\,", "\\frac{d}{dx} P_{n+1}(x) = (2n+1) P_n(x) + \\bigl(2(n-2)+1\\bigr) P_{n-2}(x) + \\bigl(2(n-4)+1\\bigr) P_{n-4}(x) + \\cdots", "\\frac{d}{dx} P_{n+1}(x) = \\frac{2 P_n(x)}{\\left\\| P_n \\right\\|^2} + \\frac{2 P_{n-2}(x)}{\\left\\| P_{n-2} \\right\\|^2} + \\cdots", "\\| P_n \\| = \\sqrt{\\int_{-1}^1 \\bigl(P_n(x)\\bigr)^2 \\,dx} = \\sqrt{\\frac{2}{2 n + 1}} \\,", "\\begin{align}P_\\ell (\\cos \\theta) &= \\sqrt{\\frac{\\theta}{\\sin \\theta}} \\, J_0((\\ell+1/2)\\theta) + \\mathcal{O}\\left(\\ell^{-1}\\right) \\\\&= \\frac{2}{\\sqrt{2\\pi \\ell\\sin\\theta}}\\cos\\left(\\left(\\ell + \\tfrac12\\right)\\theta - \\frac{\\pi}{4}\\right) + \\mathcal{O}\\left(\\ell^{-3/2}\\right), \\quad \\theta \\in (0,\\pi),\\end{align}", "\\begin{align}P_\\ell \\left(\\frac{1}{\\sqrt{1-e^2}}\\right) &= I_0(\\ell e) + \\mathcal{O}\\left(\\ell^{-1}\\right) \\\\&= \\frac{1}{\\sqrt{2\\pi\\ell e}} \\frac{(1+e)^\\frac{\\ell+1}{2}}{(1-e)^\\frac{\\ell}{2}} + \\mathcal{O}\\left(\\ell^{-1}\\right)\\,,\\end{align}", "P_n(\\pm 1) \\ne 0", "dP_n(x)/dx", "P_n(1) = 1 \\,, \\quad P_n(-1) = \\begin{cases} 1 & \\text{for} \\quad n = 2m \\\\ -1 & \\text{for} \\quad n = 2m+1 \\,. \\end{cases}", "P_n(0) = \\begin{cases} \\frac{(-1)^{m}}{4^m} \\tbinom{2m}{m} = \\frac{(-1)^{m}}{2^{2m}} \\frac{(2m)!}{\\left(m!\\right)^2} & \\text{for} \\quad n = 2m \\\\ 0 & \\text{for} \\quad n = 2m+1 \\,. \\end{cases}", "P\\~_{n}(x)", "R_n(x) = \\frac{\\sqrt{2}}{x+1}\\,P_n\\left(\\frac{x-1}{x+1}\\right)\\," ],
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}