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 \mathrm{H}_n(x)}

${\displaystyle \mathrm {H} _{n}(x)}$

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

Semantic latex: \StruveH{n}@{x}

Confidence: 0.6805

Mathematica

Translation: StruveH[n, x]

Information

Sub Equations

StruveH[n, x]

Free variables

n

x

Symbol info

Struve function; Example: \StruveH{\nu}@{z}

Will be translated to: StruveH[$0, $1] Relevant links to definitions: DLMF: http://dlmf.nist.gov/11.2#E1 Mathematica: https://reference.wolfram.com/language/ref/StruveH.html

Tests

Symbolic

Numeric

SymPy

Translation:

Information

Symbol info

(LaTeX -> SymPy) No translation possible for given token: Cannot extract information from feature set: \StruveH [\StruveH]

Tests

Symbolic

Numeric

Maple

Translation: StruveH(n, x)

Information

Sub Equations

StruveH(n, x)

Free variables

n

x

Symbol info

Struve function; Example: \StruveH{\nu}@{z}

Will be translated to: StruveH($0, $1) Relevant links to definitions: DLMF: http://dlmf.nist.gov/11.2#E1 Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=StruveH

Tests

Symbolic

Numeric

Dependency Graph Information

Includes

$\mathbf {E} _{n}$

$n$

$x$

Is part of

${\begin{aligned}\mathbf {E} _{n}(z)&={\frac {1}{\pi }}\sum _{k=0}^{\left\lfloor {\frac {n-1}{2}}\right\rfloor }{\frac {\Gamma \left(k+{\frac {1}{2}}\right)\left({\frac {z}{2}}\right)^{n-2k-1}}{\Gamma \left(n-k+{\frac {1}{2}}\right)}}-\mathbf {H} _{n}(z),\\\mathbf {E} _{-n}(z)&={\frac {(-1)^{n+1}}{\pi }}\sum _{k=0}^{\left\lfloor {\frac {n-1}{2}}\right\rfloor }{\frac {\Gamma (n-k-{\frac {1}{2}})\left({\frac {z}{2}}\right)^{-n+2k+1}}{\Gamma \left(k+{\frac {3}{2}}\right)}}-\mathbf {H} _{-n}(z).\end{aligned}}$

Complete translation information:

{
  "id" : "FORMULA_6de914ff4efc3e766010f298f87691b1",
  "formula" : "\\mathrm{H}_n(x)",
  "semanticFormula" : "\\StruveH{n}@{x}",
  "confidence" : 0.6805,
  "translations" : {
    "Mathematica" : {
      "translation" : "StruveH[n, x]",
      "translationInformation" : {
        "subEquations" : [ "StruveH[n, x]" ],
        "freeVariables" : [ "n", "x" ],
        "tokenTranslations" : {
          "\\StruveH" : "Struve function; Example: \\StruveH{\\nu}@{z}\nWill be translated to: StruveH[$0, $1]\nRelevant links to definitions:\nDLMF:         http://dlmf.nist.gov/11.2#E1\nMathematica:  https://reference.wolfram.com/language/ref/StruveH.html"
        }
      },
      "numericResults" : {
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        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 0,
        "wasAborted" : false,
        "crashed" : false,
        "testCalculationsGroups" : [ ]
      },
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        "crashed" : false,
        "testCalculationsGroup" : [ ]
      }
    },
    "SymPy" : {
      "translation" : "",
      "translationInformation" : {
        "tokenTranslations" : {
          "Error" : "(LaTeX -> SymPy) No translation possible for given token: Cannot extract information from feature set: \\StruveH [\\StruveH]"
        }
      },
      "numericResults" : {
        "overallResult" : "SKIPPED",
        "numberOfTests" : 0,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 0,
        "wasAborted" : false,
        "crashed" : false,
        "testCalculationsGroups" : [ ]
      },
      "symbolicResults" : {
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        "testCalculationsGroup" : [ ]
      }
    },
    "Maple" : {
      "translation" : "StruveH(n, x)",
      "translationInformation" : {
        "subEquations" : [ "StruveH(n, x)" ],
        "freeVariables" : [ "n", "x" ],
        "tokenTranslations" : {
          "\\StruveH" : "Struve function; Example: \\StruveH{\\nu}@{z}\nWill be translated to: StruveH($0, $1)\nRelevant links to definitions:\nDLMF:  http://dlmf.nist.gov/11.2#E1\nMaple: https://www.maplesoft.com/support/help/maple/view.aspx?path=StruveH"
        }
      },
      "numericResults" : {
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        "numberOfTests" : 0,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
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        "numberOfErrorTests" : 0,
        "wasAborted" : false,
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        "testCalculationsGroups" : [ ]
      },
      "symbolicResults" : {
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        "numberOfTests" : 0,
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        "numberOfErrorTests" : 0,
        "crashed" : false,
        "testCalculationsGroup" : [ ]
      }
    }
  },
  "positions" : [ ],
  "includes" : [ "\\mathbf{E}_{n}", "n", "x" ],
  "isPartOf" : [ "\\begin{align}\\mathbf{E}_n(z)    &= \\frac{1}{\\pi} \\sum_{k=0}^{\\left \\lfloor \\frac{n-1}{2} \\right \\rfloor} \\frac{\\Gamma \\left (k+ \\frac{1}{2} \\right) \\left (\\frac{z}{2} \\right )^{n-2k-1}}{\\Gamma \\left (n- k + \\frac{1}{2}\\right )} -\\mathbf{H}_n(z),\\\\\\mathbf{E}_{-n}(z) &= \\frac{(-1)^{n+1}}{\\pi}\\sum_{k=0}^{\\left \\lfloor \\frac{n-1}{2} \\right \\rfloor} \\frac{\\Gamma(n-k-\\frac{1}{2}) \\left (\\frac{z}{2} \\right )^{-n+2k+1}}{\\Gamma \\left (k+ \\frac{3}{2} \\right)}-\\mathbf{H}_{-n}(z). \\end{align}" ],
  "definiens" : [ ]
}

Specify your own input

Formula input
Wikitext context	[[File:Mplwp Struve function05.svg\|320px\|thumb\|Graph of <math>\mathrm{H}_n(x)</math> for <math>n\in [0,1,2,3,4,5]</math>]] In [[mathematics]], the '''Struve functions''' {{math\|'''H'''<sub>''α''</sub>(''x'')}}, are solutions {{math\|''y''(''x'')}} of the non-homogeneous [[Bessel's differential equation]]: : <math>x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + \left (x^2 - \alpha^2 \right )y = \frac{4\left (\frac{x}{2}\right)^{\alpha+1}}{\sqrt{\pi}\Gamma \left (\alpha+\frac{1}{2} \right )}</math> introduced by {{harvs\|txt\|last=Struve\|first=Hermann\|authorlink=Hermann Struve\|year=1882}}. The [[complex number]] α is the '''order''' of the Struve function, and is often an integer. And further defined its second-kind version <math>\mathbf{K}_\alpha(x)</math> as <math>\mathbf{K}_\alpha(x)=\mathbf{H}_\alpha(x)-Y_\alpha(x)</math>. The '''modified Struve functions''' {{math\|'''L'''<sub>''α''</sub>(''x'')}} are equal to {{math\|−''ie''<sup>−''iαπ'' / 2</sup>'''H'''<sub>''α''</sub>(''ix'')}}, are solutions {{math\|''y''(''x'')}} of the non-homogeneous [[Bessel's differential equation]]: : <math>x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} - \left (x^2 + \alpha^2 \right )y = \frac{4\left (\frac{x}{2}\right)^{\alpha+1}}{\sqrt{\pi}\Gamma \left (\alpha+\frac{1}{2} \right )}</math> And further defined its second-kind version <math>\mathbf{M}_\alpha(x)</math> as <math>\mathbf{M}_\alpha(x)=\mathbf{L}_\alpha(x)-I_\alpha(x)</math>. ==Definitions== Since this is a [[Ordinary differential equation#Nonhomogeneous equations\|non-homogeneous]] equation, solutions can be constructed from a single particular solution by adding the solutions of the homogeneous problem. In this case, the homogeneous solutions are the [[Bessel function]]s, and the particular solution may be chosen as the corresponding Struve function. ===Power series expansion=== Struve functions, denoted as {{math\|'''H'''<sub>''α''</sub>(''z'')}} have the power series form :<math> \mathbf{H}_\alpha(z) = \sum_{m=0}^\infty \frac{(-1)^m}{\Gamma \left (m+\frac{3}{2} \right ) \Gamma \left (m+\alpha+\frac{3}{2} \right )} \left({\frac{z}{2}}\right)^{2m+\alpha+1},</math> where {{math\|Γ(''z'')}} is the [[gamma function]]. The modified Struve functions, denoted {{math\|'''L'''<sub>''ν''</sub>(''z'')}}, have the following power series form :<math> \mathbf{L}_{\nu}(z) = \left({\frac{z}{2}}\right)^{\nu+1} \sum_{k=0}^\infty \frac{1}{\Gamma \left (\frac{3}{2}+k \right ) \Gamma \left (\frac{3}{2}+k+\nu \right )} \left(\frac{z}{2}\right)^{2k}.</math> ===Integral form=== Another definition of the Struve function, for values of {{mvar\|α}} satisfying {{math\|Re(''α'') > − {{sfrac\|1\|2}}}}, is possible expressing in term of the Poisson’s integral representation: :<math>\mathbf{H}_\alpha(x)=\frac{2\left(\frac{x}{2}\right)^\alpha}{\sqrt\pi\Gamma\left(\alpha+\frac{1}{2}\right)}\int_0^1(1-t^2)^{\alpha-\frac{1}{2}}\sin xt~dt=\frac{2\left(\frac{x}{2}\right)^\alpha}{\sqrt\pi\Gamma\left(\alpha+\frac{1}{2}\right)}\int_0^\frac{\pi}{2}\sin(x\cos\tau)\sin^{2\alpha}\tau~d\tau=\frac{2\left(\frac{x}{2}\right)^\alpha}{\sqrt\pi\Gamma\left(\alpha+\frac{1}{2}\right)}\int_0^\frac{\pi}{2}\sin(x\sin\tau)\cos^{2\alpha}\tau~d\tau</math> :<math>\mathbf{K}_\alpha(x)=\frac{2\left(\frac{x}{2}\right)^\alpha}{\sqrt\pi\Gamma\left(\alpha+\frac{1}{2}\right)}\int_0^\infty(1+t^2)^{\alpha-\frac{1}{2}}e^{-xt}~dt=\frac{2\left(\frac{x}{2}\right)^\alpha}{\sqrt\pi\Gamma\left(\alpha+\frac{1}{2}\right)}\int_0^\infty e^{-x\sinh\tau}\cosh^{2\alpha}\tau~d\tau</math> :<math>\mathbf{L}_\alpha(x)=\frac{2\left(\frac{x}{2}\right)^\alpha}{\sqrt\pi\Gamma\left(\alpha+\frac{1}{2}\right)}\int_0^1(1-t^2)^{\alpha-\frac{1}{2}}\sinh xt~dt=\frac{2\left(\frac{x}{2}\right)^\alpha}{\sqrt\pi\Gamma\left(\alpha+\frac{1}{2}\right)}\int_0^\frac{\pi}{2}\sinh(x\cos\tau)\sin^{2\alpha}\tau~d\tau=\frac{2\left(\frac{x}{2}\right)^\alpha}{\sqrt\pi\Gamma\left(\alpha+\frac{1}{2}\right)}\int_0^\frac{\pi}{2}\sinh(x\sin\tau)\cos^{2\alpha}\tau~d\tau</math> :<math>\mathbf{M}_\alpha(x)=-\frac{2\left(\frac{x}{2}\right)^\alpha}{\sqrt\pi\Gamma\left(\alpha+\frac{1}{2}\right)}\int_0^1(1-t^2)^{\alpha-\frac{1}{2}}e^{-xt}~dt=-\frac{2\left(\frac{x}{2}\right)^\alpha}{\sqrt\pi\Gamma\left(\alpha+\frac{1}{2}\right)}\int_0^\frac{\pi}{2}e^{-x\cos\tau}\sin^{2\alpha}\tau~d\tau=-\frac{2\left(\frac{x}{2}\right)^\alpha}{\sqrt\pi\Gamma\left(\alpha+\frac{1}{2}\right)}\int_0^\frac{\pi}{2}e^{-x\sin\tau}\cos^{2\alpha}\tau~d\tau</math> ==Asymptotic forms== For small {{mvar\|x}}, the power series expansion is given [[#Power series expansion\|above]]. For large {{mvar\|x}}, one obtains: :<math>\mathbf{H}_\alpha(x) - Y_\alpha(x) = \frac{\left(\frac{x}{2}\right)^{\alpha-1}}{\sqrt{\pi} \Gamma \left (\alpha+\frac{1}{2} \right )} + O\left(\left (\tfrac{x}{2}\right)^{\alpha-3}\right),</math> where {{math\|''Y<sub>α</sub>''(''x'')}} is the [[Bessel function#Bessel functions of the second kind : Y.CE.B1\|Neumann function]]. ==Properties== The Struve functions satisfy the following recurrence relations: :<math>\begin{align} \mathbf{H}_{\alpha -1}(x) + \mathbf{H}_{\alpha+1}(x) &= \frac{2\alpha}{x} \mathbf{H}_\alpha (x) + \frac{\left (\frac{x}{2}\right)^{\alpha}}{\sqrt{\pi}\Gamma \left (\alpha + \frac{3}{2} \right )}, \\ \mathbf{H}_{\alpha -1}(x) - \mathbf{H}_{\alpha+1}(x) &= 2 \frac{d}{dx} \left (\mathbf{H}_\alpha(x) \right) - \frac{ \left( \frac{x}{2} \right)^\alpha}{\sqrt{\pi}\Gamma \left (\alpha + \frac{3}{2} \right )}. \end{align}</math> ==Relation to other functions== Struve functions of integer order can be expressed in terms of [[Anger function\|Weber function]]s {{math\|'''E'''<sub>''n''</sub>}} and vice versa: if {{mvar\|n}} is a non-negative integer then :<math>\begin{align} \mathbf{E}_n(z) &= \frac{1}{\pi} \sum_{k=0}^{\left \lfloor \frac{n-1}{2} \right \rfloor} \frac{\Gamma \left (k+ \frac{1}{2} \right) \left (\frac{z}{2} \right )^{n-2k-1}}{\Gamma \left (n- k + \frac{1}{2}\right )} -\mathbf{H}_n(z),\\ \mathbf{E}_{-n}(z) &= \frac{(-1)^{n+1}}{\pi}\sum_{k=0}^{\left \lfloor \frac{n-1}{2} \right \rfloor} \frac{\Gamma(n-k-\frac{1}{2}) \left (\frac{z}{2} \right )^{-n+2k+1}}{\Gamma \left (k+ \frac{3}{2} \right)}-\mathbf{H}_{-n}(z). \end{align}</math> Struve functions of order {{math\|''n'' + {{sfrac\|1\|2}}}} where {{mvar\|n}} is an integer can be expressed in terms of elementary functions. In particular if {{mvar\|n}} is a non-negative integer then :<math>\mathbf{H}_{-n-\frac{1}{2}} (z) = (-1)^n J_{n+\frac{1}{2}}(z),</math> where the right hand side is a [[spherical Bessel function]]. Struve functions (of any order) can be expressed in terms of the [[generalized hypergeometric function]] {{math\|<sub>1</sub>''F''<sub>2</sub>}} (which is '''not''' the Gauss hypergeometric function {{math\|<sub>2</sub>''F''<sub>1</sub>}}): :<math>\mathbf{H}_{\alpha}(z) = \frac{z^{\alpha+1}}{2^{\alpha}\sqrt{\pi} \Gamma \left (\alpha+\tfrac{3}{2} \right )} {}_1F_2 \left (1,\tfrac{3}{2}, \alpha+\tfrac{3}{2},-\tfrac{z^2}{4} \right ).</math> ==References== {{cite journal\|doi=10.1121/1.1564019 \|author=R. M. Aarts and Augustus J. E. M. Janssen \|title=Approximation of the Struve function ''H''<sub>1</sub> occurring in impedance calculations \|journal= J. Acoust. Soc. Am. \|volume= 113 \|pages= 2635–2637 \|year= 2003 \|pmid=12765381 \|issue=5 \|bibcode = 2003ASAJ..113.2635A }} {{cite journal\|doi=10.1121/1.4968792 \|pmid=28040027 \|author=R. M. Aarts and Augustus J. E. M. Janssen \|title=Efficient approximation of the Struve functions ''H''<sub>''n''</sub> occurring in the calculation of sound radiation quantities \|journal= J. Acoust. Soc. Am. \|volume= 140 \|pages= 4154–4160 \|year= 2016 \|issue=6 \|bibcode=2016ASAJ..140.4154A\|url=https://research.tue.nl/nl/publications/efficient-approximation-of-the-struve-functions-hn-occurring-in-the-calculation-of-sound-radiation-quantaties(c68b8858-9c9d-4ff2-bf39-e888bb638527).html }} {{AS ref\|12\|496}} {{springer\|id=S/s090700\|first=A. B. \|last=Ivanov}} {{dlmf\|id=11\|Struve and Related Functions\|first=R. B. \|last=Paris}} {{cite journal\|doi=10.1002/andp.18822531319 \|first=H. \|last=Struve \|title=Beitrag zur Theorie der Diffraction an Fernröhren \|journal= Annalen der Physik und Chemie \|volume= 17\|issue=13 \|year=1882 \|pages= 1008–1016\|bibcode = 1882AnP...253.1008S \|url=https://zenodo.org/record/1423790 }} ==External links== *[http://functions.wolfram.com/Bessel-TypeFunctions/StruveH/introductions/Struves/ Struve functions] at [http://functions.wolfram.com the Wolfram functions site]. {{DEFAULTSORT:Struve Function}} [[Category:Special functions]] [[Category:Struve family]]
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LaTeX to CAS translator

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