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 \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}}

${\displaystyle {\begin{aligned}\mathbf {H} _{\alpha -1}(x)+\mathbf {H} _{\alpha +1}(x)&={\frac {2\alpha }{x}}\mathbf {H} _{\alpha }(x)+{\frac {\left({\frac {x}{2}}\right)^{\alpha }}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {3}{2}}\right)}},\\\mathbf {H} _{\alpha -1}(x)-\mathbf {H} _{\alpha +1}(x)&=2{\frac {d}{dx}}\left(\mathbf {H} _{\alpha }(x)\right)-{\frac {\left({\frac {x}{2}}\right)^{\alpha }}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {3}{2}}\right)}}.\end{aligned}}}$

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

Semantic latex: \begin{align}\StruveH{\alpha -1}@{x} + \StruveH{\alpha+1}@{x} &= \frac{2\alpha}{x} \StruveH{\alpha}@{x} + \frac{(\frac{x}{2})^{\alpha}}{\sqrt{\cpi} \EulerGamma@{\alpha + \frac{3}{2}}} , \\ \StruveH{\alpha -1}@{x} - \StruveH{\alpha+1}@{x} &= 2 \deriv [1]{ }{x}(\StruveH{\alpha}@{x}) - \frac{(\frac{x}{2})^\alpha{\sqrt{\cpi} \EulerGamma@{\alpha + \frac{3}{2}}}} .\end{align}

Confidence: 0.66012076646182

Mathematica

Translation: StruveH[\[Alpha]- 1, x]+ StruveH[\[Alpha]+ 1, x] == Divide[2*\[Alpha],x]*StruveH[\[Alpha], x]+Divide[(Divide[x,2])^\[Alpha],Sqrt[Pi]*Gamma[\[Alpha]+Divide[3,2]]] StruveH[\[Alpha]- 1, x]- StruveH[\[Alpha]+ 1, x] == 2*D[(StruveH[\[Alpha], x])-Divide[(Divide[x,2])^\[Alpha]*Sqrt[Pi]*Gamma[\[Alpha]+Divide[3,2]],], {x, 1}]

Information

Sub Equations

StruveH[\[Alpha]- 1, x]+ StruveH[\[Alpha]+ 1, x] = Divide[2*\[Alpha],x]*StruveH[\[Alpha], x]+Divide[(Divide[x,2])^\[Alpha],Sqrt[Pi]*Gamma[\[Alpha]+Divide[3,2]]]

StruveH[\[Alpha]- 1, x]- StruveH[\[Alpha]+ 1, x] = 2*D[(StruveH[\[Alpha], x])-Divide[(Divide[x,2])^\[Alpha]*Sqrt[Pi]*Gamma[\[Alpha]+Divide[3,2]],], {x, 1}]

Free variables

\[Alpha]

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

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

Pi was translated to: Pi

Could be the second Feigenbaum constant.

But this system doesn't know how to translate it as a constant. It was translated as a general letter.

Euler Gamma function; Example: \EulerGamma@{z}

Will be translated to: Gamma[$0] Relevant links to definitions: DLMF: http://dlmf.nist.gov/5.2#E1 Mathematica: https://reference.wolfram.com/language/ref/Gamma.html

Tests

Symbolic

Test expression: (StruveH[\[Alpha]- 1, x]+ StruveH[\[Alpha]+ 1, x])-(Divide[2*\[Alpha],x]*StruveH[\[Alpha], x]+Divide[(Divide[x,2])^\[Alpha],Sqrt[Pi]*Gamma[\[Alpha]+Divide[3,2]]])

ERROR:

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

Test expression: (StruveH[\[Alpha]- 1, x]- StruveH[\[Alpha]+ 1, x])-(2*D[(StruveH[\[Alpha], x])-Divide[(Divide[x,2])^\[Alpha]*Sqrt[Pi]*Gamma[\[Alpha]+Divide[3,2]],], {x, 1}])

ERROR:

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

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(alpha - 1, x)+ StruveH(alpha + 1, x) = (2*alpha)/(x)*StruveH(alpha, x)+(((x)/(2))^(alpha))/(sqrt(Pi)*GAMMA(alpha +(3)/(2))); StruveH(alpha - 1, x)- StruveH(alpha + 1, x) = 2*diff((StruveH(alpha, x))-(((x)/(2))^(alpha)*sqrt(Pi)*GAMMA(alpha +(3)/(2)))/(), [x$(1)])

Information

Sub Equations

StruveH(alpha - 1, x)+ StruveH(alpha + 1, x) = (2*alpha)/(x)*StruveH(alpha, x)+(((x)/(2))^(alpha))/(sqrt(Pi)*GAMMA(alpha +(3)/(2)))

StruveH(alpha - 1, x)- StruveH(alpha + 1, x) = 2*diff((StruveH(alpha, x))-(((x)/(2))^(alpha)*sqrt(Pi)*GAMMA(alpha +(3)/(2)))/(), [x$(1)])

Free variables

alpha

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

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

Pi was translated to: Pi

Could be the second Feigenbaum constant.

But this system doesn't know how to translate it as a constant. It was translated as a general letter.

Euler Gamma function; Example: \EulerGamma@{z}

Will be translated to: GAMMA($0) Relevant links to definitions: DLMF: http://dlmf.nist.gov/5.2#E1 Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=GAMMA

Tests

Symbolic

Numeric

Dependency Graph Information

Includes

$\mathbf {K} _{\alpha }(x)$

$\alpha$

$\Gamma (z)$

$\mathbf {H} _{\alpha }(x)$

$\mathbf {L} _{\alpha }(x)$

$x$

$\mathbf {H} _{\alpha }(z)$

$Y_{\alpha }(x)$

$\mathbf {M} _{\alpha }(x)$

${\begin{aligned}\mathbf {H} _{\alpha -1}(x)+\mathbf {H} _{\alpha +1}(x)&={\frac {2\alpha }{x}}\mathbf {H} _{\alpha }(x)+{\frac {\left({\frac {x}{2}}\right)^{\alpha }}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {3}{2}}\right)}},\\\mathbf {H} _{\alpha -1}(x)-\mathbf {H} _{\alpha +1}(x)&=2{\frac {d}{dx}}\left(\mathbf {H} _{\alpha }(x)\right)-{\frac {\left({\frac {x}{2}}\right)^{\alpha }}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {3}{2}}\right)}}.\end{aligned}}$

Is part of

${\begin{aligned}\mathbf {H} _{\alpha -1}(x)+\mathbf {H} _{\alpha +1}(x)&={\frac {2\alpha }{x}}\mathbf {H} _{\alpha }(x)+{\frac {\left({\frac {x}{2}}\right)^{\alpha }}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {3}{2}}\right)}},\\\mathbf {H} _{\alpha -1}(x)-\mathbf {H} _{\alpha +1}(x)&=2{\frac {d}{dx}}\left(\mathbf {H} _{\alpha }(x)\right)-{\frac {\left({\frac {x}{2}}\right)^{\alpha }}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {3}{2}}\right)}}.\end{aligned}}$

Complete translation information:

{
  "id" : "FORMULA_bb3a0a81a718747763efe1de1aaebfc8",
  "formula" : "\\begin{align}\n\\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 )}, \\\\\n\\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 )}.\n\\end{align}",
  "semanticFormula" : "\\begin{align}\\StruveH{\\alpha -1}@{x} + \\StruveH{\\alpha+1}@{x} &= \\frac{2\\alpha}{x} \\StruveH{\\alpha}@{x} + \\frac{(\\frac{x}{2})^{\\alpha}}{\\sqrt{\\cpi} \\EulerGamma@{\\alpha + \\frac{3}{2}}} , \\\\ \\StruveH{\\alpha -1}@{x} - \\StruveH{\\alpha+1}@{x} &= 2 \\deriv [1]{ }{x}(\\StruveH{\\alpha}@{x}) - \\frac{(\\frac{x}{2})^\\alpha{\\sqrt{\\cpi} \\EulerGamma@{\\alpha + \\frac{3}{2}}}} .\\end{align}",
  "confidence" : 0.6601207664618238,
  "translations" : {
    "Mathematica" : {
      "translation" : "StruveH[\\[Alpha]- 1, x]+ StruveH[\\[Alpha]+ 1, x] == Divide[2*\\[Alpha],x]*StruveH[\\[Alpha], x]+Divide[(Divide[x,2])^\\[Alpha],Sqrt[Pi]*Gamma[\\[Alpha]+Divide[3,2]]]\nStruveH[\\[Alpha]- 1, x]- StruveH[\\[Alpha]+ 1, x] == 2*D[(StruveH[\\[Alpha], x])-Divide[(Divide[x,2])^\\[Alpha]*Sqrt[Pi]*Gamma[\\[Alpha]+Divide[3,2]],], {x, 1}]",
      "translationInformation" : {
        "subEquations" : [ "StruveH[\\[Alpha]- 1, x]+ StruveH[\\[Alpha]+ 1, x] = Divide[2*\\[Alpha],x]*StruveH[\\[Alpha], x]+Divide[(Divide[x,2])^\\[Alpha],Sqrt[Pi]*Gamma[\\[Alpha]+Divide[3,2]]]", "StruveH[\\[Alpha]- 1, x]- StruveH[\\[Alpha]+ 1, x] = 2*D[(StruveH[\\[Alpha], x])-Divide[(Divide[x,2])^\\[Alpha]*Sqrt[Pi]*Gamma[\\[Alpha]+Divide[3,2]],], {x, 1}]" ],
        "freeVariables" : [ "\\[Alpha]", "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",
          "\\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",
          "\\cpi" : "Pi was translated to: Pi",
          "\\alpha" : "Could be the second Feigenbaum constant.\nBut this system doesn't know how to translate it as a constant. It was translated as a general letter.\n",
          "\\EulerGamma" : "Euler Gamma function; Example: \\EulerGamma@{z}\nWill be translated to: Gamma[$0]\nRelevant links to definitions:\nDLMF:         http://dlmf.nist.gov/5.2#E1\nMathematica:  https://reference.wolfram.com/language/ref/Gamma.html"
        }
      },
      "numericResults" : {
        "overallResult" : "SKIPPED",
        "numberOfTests" : 0,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 0,
        "wasAborted" : false,
        "crashed" : false,
        "testCalculationsGroups" : [ ]
      },
      "symbolicResults" : {
        "overallResult" : "ERROR",
        "numberOfTests" : 2,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 2,
        "crashed" : false,
        "testCalculationsGroup" : [ {
          "lhs" : "StruveH[\\[Alpha]- 1, x]+ StruveH[\\[Alpha]+ 1, x]",
          "rhs" : "Divide[2*\\[Alpha],x]*StruveH[\\[Alpha], x]+Divide[(Divide[x,2])^\\[Alpha],Sqrt[Pi]*Gamma[\\[Alpha]+Divide[3,2]]]",
          "testExpression" : "(StruveH[\\[Alpha]- 1, x]+ StruveH[\\[Alpha]+ 1, x])-(Divide[2*\\[Alpha],x]*StruveH[\\[Alpha], x]+Divide[(Divide[x,2])^\\[Alpha],Sqrt[Pi]*Gamma[\\[Alpha]+Divide[3,2]]])",
          "testCalculations" : [ {
            "result" : "ERROR",
            "testTitle" : "Simple",
            "testExpression" : null,
            "resultExpression" : null,
            "wasAborted" : false,
            "conditionallySuccessful" : false
          } ]
        }, {
          "lhs" : "StruveH[\\[Alpha]- 1, x]- StruveH[\\[Alpha]+ 1, x]",
          "rhs" : "2*D[(StruveH[\\[Alpha], x])-Divide[(Divide[x,2])^\\[Alpha]*Sqrt[Pi]*Gamma[\\[Alpha]+Divide[3,2]],], {x, 1}]",
          "testExpression" : "(StruveH[\\[Alpha]- 1, x]- StruveH[\\[Alpha]+ 1, x])-(2*D[(StruveH[\\[Alpha], x])-Divide[(Divide[x,2])^\\[Alpha]*Sqrt[Pi]*Gamma[\\[Alpha]+Divide[3,2]],], {x, 1}])",
          "testCalculations" : [ {
            "result" : "ERROR",
            "testTitle" : "Simple",
            "testExpression" : null,
            "resultExpression" : null,
            "wasAborted" : false,
            "conditionallySuccessful" : false
          } ]
        } ]
      }
    },
    "SymPy" : {
      "translation" : "",
      "translationInformation" : {
        "tokenTranslations" : {
          "Error" : "(LaTeX -> SymPy) No translation possible for given token: Cannot extract information from feature set: \\StruveH [\\StruveH]"
        }
      }
    },
    "Maple" : {
      "translation" : "StruveH(alpha - 1, x)+ StruveH(alpha + 1, x) = (2*alpha)/(x)*StruveH(alpha, x)+(((x)/(2))^(alpha))/(sqrt(Pi)*GAMMA(alpha +(3)/(2))); StruveH(alpha - 1, x)- StruveH(alpha + 1, x) = 2*diff((StruveH(alpha, x))-(((x)/(2))^(alpha)*sqrt(Pi)*GAMMA(alpha +(3)/(2)))/(), [x$(1)])",
      "translationInformation" : {
        "subEquations" : [ "StruveH(alpha - 1, x)+ StruveH(alpha + 1, x) = (2*alpha)/(x)*StruveH(alpha, x)+(((x)/(2))^(alpha))/(sqrt(Pi)*GAMMA(alpha +(3)/(2)))", "StruveH(alpha - 1, x)- StruveH(alpha + 1, x) = 2*diff((StruveH(alpha, x))-(((x)/(2))^(alpha)*sqrt(Pi)*GAMMA(alpha +(3)/(2)))/(), [x$(1)])" ],
        "freeVariables" : [ "alpha", "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",
          "\\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",
          "\\cpi" : "Pi was translated to: Pi",
          "\\alpha" : "Could be the second Feigenbaum constant.\nBut this system doesn't know how to translate it as a constant. It was translated as a general letter.\n",
          "\\EulerGamma" : "Euler Gamma function; Example: \\EulerGamma@{z}\nWill be translated to: GAMMA($0)\nRelevant links to definitions:\nDLMF:  http://dlmf.nist.gov/5.2#E1\nMaple: https://www.maplesoft.com/support/help/maple/view.aspx?path=GAMMA"
        }
      }
    }
  },
  "positions" : [ ],
  "includes" : [ "\\mathbf{K}_\\alpha(x)", "\\alpha", "\\Gamma(z)", "\\mathbf{H}_{\\alpha}(x)", "\\mathbf{L}_{\\alpha}(x)", "x", "\\mathbf{H}_{\\alpha}(z)", "Y_{\\alpha}(x)", "\\mathbf{M}_\\alpha(x)", "\\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}" ],
  "isPartOf" : [ "\\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}" ],
  "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]]
	Purge cached results.

LaTeX to CAS translator

Mathematica

Information

Tests

Symbolic

Numeric

SymPy

Information

Tests

Symbolic

Numeric

Maple

Information

Tests

Symbolic

Numeric

Dependency Graph Information

Specify your own input

Navigation menu

Search