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 \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}

${\displaystyle \mathbf {K} _{\alpha }(x)={\frac {2\left({\frac {x}{2}}\right)^{\alpha }}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {1}{2}}\right)}}\int _{0}^{\infty }(1+t^{2})^{\alpha -{\frac {1}{2}}}e^{-xt}~dt={\frac {2\left({\frac {x}{2}}\right)^{\alpha }}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {1}{2}}\right)}}\int _{0}^{\infty }e^{-x\sinh \tau }\cosh ^{2\alpha }\tau ~d\tau }$

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

Semantic latex: \StruveK{\alpha}@{x} = \frac{2(\frac{x}{2})^\alpha}{\sqrt{\cpi} \EulerGamma@{\alpha + \frac{1}{2}}} \int_0^\infty(1 + t^2)^{\alpha-\frac{1}{2}} \expe^{-xt} \diff{t} = \frac{2(\frac{x}{2})^\alpha}{\sqrt{\cpi} \EulerGamma@{\alpha + \frac{1}{2}}} \int_0^\infty \expe^{-x\sinh\tau} \cosh^{2\alpha} \tau \diff{\tau}

Confidence: 0.66012076646182

Mathematica

Translation: StruveH[\[Alpha], x] - BesselY[\[Alpha], x] == Divide[2*(Divide[x,2])^\[Alpha],Sqrt[Pi]*Gamma[\[Alpha]+Divide[1,2]]]*Integrate[(1 + (t)^(2))^(\[Alpha]-Divide[1,2])* Exp[- x*t], {t, 0, Infinity}, GenerateConditions->None] == Divide[2*(Divide[x,2])^\[Alpha],Sqrt[Pi]*Gamma[\[Alpha]+Divide[1,2]]]*Integrate[Exp[- x*Sinh[\[Tau]]]*(Cosh[\[Tau]])^(2*\[Alpha]), {\[Tau], 0, Infinity}, GenerateConditions->None]

Information

Sub Equations

StruveH[\[Alpha], x] - BesselY[\[Alpha], x] = Divide[2*(Divide[x,2])^\[Alpha],Sqrt[Pi]*Gamma[\[Alpha]+Divide[1,2]]]*Integrate[(1 + (t)^(2))^(\[Alpha]-Divide[1,2])* Exp[- x*t], {t, 0, Infinity}, GenerateConditions->None]

Divide[2*(Divide[x,2])^\[Alpha],Sqrt[Pi]*Gamma[\[Alpha]+Divide[1,2]]]*Integrate[(1 + (t)^(2))^(\[Alpha]-Divide[1,2])* Exp[- x*t], {t, 0, Infinity}, GenerateConditions->None] = Divide[2*(Divide[x,2])^\[Alpha],Sqrt[Pi]*Gamma[\[Alpha]+Divide[1,2]]]*Integrate[Exp[- x*Sinh[\[Tau]]]*(Cosh[\[Tau]])^(2*\[Alpha]), {\[Tau], 0, Infinity}, GenerateConditions->None]

Free variables

\[Alpha]

x

Symbol info

Hyperbolic cosine; Example: \cosh@@{z}

Will be translated to: Cosh[$0] Relevant links to definitions: DLMF: http://dlmf.nist.gov/4.28#E2 Mathematica: https://reference.wolfram.com/language/ref/Cosh.html

Recognizes e with power as the exponential function. It was translated as a function.

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.

Hyperbolic sine; Example: \sinh@@{z}

Will be translated to: Sinh[$0] Relevant links to definitions: DLMF: http://dlmf.nist.gov/4.28#E1 Mathematica: https://reference.wolfram.com/language/ref/Sinh.html

Pi was translated to: Pi

Associated Struve funtion; Example: \StruveK{\nu}@{z}

Will be translated to: StruveH[$0, $1] - BesselY[$0, $1] Relevant links to definitions: DLMF: http://dlmf.nist.gov/11.2#E5 Mathematica: https://reference.wolfram.com/language/ref/StruveH.html

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], x] - BesselY[\[Alpha], x])-(Divide[2*(Divide[x,2])^\[Alpha],Sqrt[Pi]*Gamma[\[Alpha]+Divide[1,2]]]*Integrate[(1 + (t)^(2))^(\[Alpha]-Divide[1,2])* Exp[- x*t], {t, 0, Infinity}, GenerateConditions->None])

ERROR:

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

Test expression: (Divide[2*(Divide[x,2])^\[Alpha],Sqrt[Pi]*Gamma[\[Alpha]+Divide[1,2]]]*Integrate[(1 + (t)^(2))^(\[Alpha]-Divide[1,2])* Exp[- x*t], {t, 0, Infinity}, GenerateConditions->None])-(Divide[2*(Divide[x,2])^\[Alpha],Sqrt[Pi]*Gamma[\[Alpha]+Divide[1,2]]]*Integrate[Exp[- x*Sinh[\[Tau]]]*(Cosh[\[Tau]])^(2*\[Alpha]), {\[Tau], 0, Infinity}, GenerateConditions->None])

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: \StruveK [\StruveK]

Tests

Symbolic

Numeric

Maple

Translation: StruveH(alpha, x) - BesselY(alpha, x) = (2*((x)/(2))^(alpha))/(sqrt(Pi)*GAMMA(alpha +(1)/(2)))*int((1 + (t)^(2))^(alpha -(1)/(2))* exp(- x*t), t = 0..infinity) = (2*((x)/(2))^(alpha))/(sqrt(Pi)*GAMMA(alpha +(1)/(2)))*int(exp(- x*sinh(tau))*(cosh(tau))^(2*alpha), tau = 0..infinity)

Information

Sub Equations

StruveH(alpha, x) - BesselY(alpha, x) = (2*((x)/(2))^(alpha))/(sqrt(Pi)*GAMMA(alpha +(1)/(2)))*int((1 + (t)^(2))^(alpha -(1)/(2))* exp(- x*t), t = 0..infinity)

(2*((x)/(2))^(alpha))/(sqrt(Pi)*GAMMA(alpha +(1)/(2)))*int((1 + (t)^(2))^(alpha -(1)/(2))* exp(- x*t), t = 0..infinity) = (2*((x)/(2))^(alpha))/(sqrt(Pi)*GAMMA(alpha +(1)/(2)))*int(exp(- x*sinh(tau))*(cosh(tau))^(2*alpha), tau = 0..infinity)

Free variables

alpha

x

Symbol info

Hyperbolic cosine; Example: \cosh@@{z}

Will be translated to: cosh($0) Relevant links to definitions: DLMF: http://dlmf.nist.gov/4.28#E2 Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=cosh

Recognizes e with power as the exponential function. It was translated as a function.

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.

Hyperbolic sine; Example: \sinh@@{z}

Will be translated to: sinh($0) Relevant links to definitions: DLMF: http://dlmf.nist.gov/4.28#E1 Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=sinh

Pi was translated to: Pi

Associated Struve funtion; Example: \StruveK{\nu}@{z}

Will be translated to: StruveH($0, $1) - BesselY($0, $1) Relevant links to definitions: DLMF: http://dlmf.nist.gov/11.2#E5 Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=StruveH

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

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

$\alpha$

$\Gamma (z)$

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

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

$x$

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

$Y_{\alpha }(x)$

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

Complete translation information:

{
  "id" : "FORMULA_d2fc4be5ffba03d935418dd9d3f2e5fc",
  "formula" : "\\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",
  "semanticFormula" : "\\StruveK{\\alpha}@{x} = \\frac{2(\\frac{x}{2})^\\alpha}{\\sqrt{\\cpi} \\EulerGamma@{\\alpha + \\frac{1}{2}}} \\int_0^\\infty(1 + t^2)^{\\alpha-\\frac{1}{2}} \\expe^{-xt} \\diff{t} = \\frac{2(\\frac{x}{2})^\\alpha}{\\sqrt{\\cpi} \\EulerGamma@{\\alpha + \\frac{1}{2}}} \\int_0^\\infty \\expe^{-x\\sinh\\tau} \\cosh^{2\\alpha} \\tau \\diff{\\tau}",
  "confidence" : 0.6601207664618238,
  "translations" : {
    "Mathematica" : {
      "translation" : "StruveH[\\[Alpha], x] - BesselY[\\[Alpha], x] == Divide[2*(Divide[x,2])^\\[Alpha],Sqrt[Pi]*Gamma[\\[Alpha]+Divide[1,2]]]*Integrate[(1 + (t)^(2))^(\\[Alpha]-Divide[1,2])* Exp[- x*t], {t, 0, Infinity}, GenerateConditions->None] == Divide[2*(Divide[x,2])^\\[Alpha],Sqrt[Pi]*Gamma[\\[Alpha]+Divide[1,2]]]*Integrate[Exp[- x*Sinh[\\[Tau]]]*(Cosh[\\[Tau]])^(2*\\[Alpha]), {\\[Tau], 0, Infinity}, GenerateConditions->None]",
      "translationInformation" : {
        "subEquations" : [ "StruveH[\\[Alpha], x] - BesselY[\\[Alpha], x] = Divide[2*(Divide[x,2])^\\[Alpha],Sqrt[Pi]*Gamma[\\[Alpha]+Divide[1,2]]]*Integrate[(1 + (t)^(2))^(\\[Alpha]-Divide[1,2])* Exp[- x*t], {t, 0, Infinity}, GenerateConditions->None]", "Divide[2*(Divide[x,2])^\\[Alpha],Sqrt[Pi]*Gamma[\\[Alpha]+Divide[1,2]]]*Integrate[(1 + (t)^(2))^(\\[Alpha]-Divide[1,2])* Exp[- x*t], {t, 0, Infinity}, GenerateConditions->None] = Divide[2*(Divide[x,2])^\\[Alpha],Sqrt[Pi]*Gamma[\\[Alpha]+Divide[1,2]]]*Integrate[Exp[- x*Sinh[\\[Tau]]]*(Cosh[\\[Tau]])^(2*\\[Alpha]), {\\[Tau], 0, Infinity}, GenerateConditions->None]" ],
        "freeVariables" : [ "\\[Alpha]", "x" ],
        "tokenTranslations" : {
          "\\cosh" : "Hyperbolic cosine; Example: \\cosh@@{z}\nWill be translated to: Cosh[$0]\nRelevant links to definitions:\nDLMF:         http://dlmf.nist.gov/4.28#E2\nMathematica:  https://reference.wolfram.com/language/ref/Cosh.html",
          "\\expe" : "Recognizes e with power as the exponential function. It was translated as a function.",
          "\\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",
          "\\sinh" : "Hyperbolic sine; Example: \\sinh@@{z}\nWill be translated to: Sinh[$0]\nRelevant links to definitions:\nDLMF:         http://dlmf.nist.gov/4.28#E1\nMathematica:  https://reference.wolfram.com/language/ref/Sinh.html",
          "\\cpi" : "Pi was translated to: Pi",
          "\\StruveK" : "Associated Struve funtion; Example: \\StruveK{\\nu}@{z}\nWill be translated to: StruveH[$0, $1] - BesselY[$0, $1]\nRelevant links to definitions:\nDLMF:         http://dlmf.nist.gov/11.2#E5\nMathematica:  https://reference.wolfram.com/language/ref/StruveH.html",
          "\\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], x] - BesselY[\\[Alpha], x]",
          "rhs" : "Divide[2*(Divide[x,2])^\\[Alpha],Sqrt[Pi]*Gamma[\\[Alpha]+Divide[1,2]]]*Integrate[(1 + (t)^(2))^(\\[Alpha]-Divide[1,2])* Exp[- x*t], {t, 0, Infinity}, GenerateConditions->None]",
          "testExpression" : "(StruveH[\\[Alpha], x] - BesselY[\\[Alpha], x])-(Divide[2*(Divide[x,2])^\\[Alpha],Sqrt[Pi]*Gamma[\\[Alpha]+Divide[1,2]]]*Integrate[(1 + (t)^(2))^(\\[Alpha]-Divide[1,2])* Exp[- x*t], {t, 0, Infinity}, GenerateConditions->None])",
          "testCalculations" : [ {
            "result" : "ERROR",
            "testTitle" : "Simple",
            "testExpression" : null,
            "resultExpression" : null,
            "wasAborted" : false,
            "conditionallySuccessful" : false
          } ]
        }, {
          "lhs" : "Divide[2*(Divide[x,2])^\\[Alpha],Sqrt[Pi]*Gamma[\\[Alpha]+Divide[1,2]]]*Integrate[(1 + (t)^(2))^(\\[Alpha]-Divide[1,2])* Exp[- x*t], {t, 0, Infinity}, GenerateConditions->None]",
          "rhs" : "Divide[2*(Divide[x,2])^\\[Alpha],Sqrt[Pi]*Gamma[\\[Alpha]+Divide[1,2]]]*Integrate[Exp[- x*Sinh[\\[Tau]]]*(Cosh[\\[Tau]])^(2*\\[Alpha]), {\\[Tau], 0, Infinity}, GenerateConditions->None]",
          "testExpression" : "(Divide[2*(Divide[x,2])^\\[Alpha],Sqrt[Pi]*Gamma[\\[Alpha]+Divide[1,2]]]*Integrate[(1 + (t)^(2))^(\\[Alpha]-Divide[1,2])* Exp[- x*t], {t, 0, Infinity}, GenerateConditions->None])-(Divide[2*(Divide[x,2])^\\[Alpha],Sqrt[Pi]*Gamma[\\[Alpha]+Divide[1,2]]]*Integrate[Exp[- x*Sinh[\\[Tau]]]*(Cosh[\\[Tau]])^(2*\\[Alpha]), {\\[Tau], 0, Infinity}, GenerateConditions->None])",
          "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: \\StruveK [\\StruveK]"
        }
      }
    },
    "Maple" : {
      "translation" : "StruveH(alpha, x) - BesselY(alpha, x) = (2*((x)/(2))^(alpha))/(sqrt(Pi)*GAMMA(alpha +(1)/(2)))*int((1 + (t)^(2))^(alpha -(1)/(2))* exp(- x*t), t = 0..infinity) = (2*((x)/(2))^(alpha))/(sqrt(Pi)*GAMMA(alpha +(1)/(2)))*int(exp(- x*sinh(tau))*(cosh(tau))^(2*alpha), tau = 0..infinity)",
      "translationInformation" : {
        "subEquations" : [ "StruveH(alpha, x) - BesselY(alpha, x) = (2*((x)/(2))^(alpha))/(sqrt(Pi)*GAMMA(alpha +(1)/(2)))*int((1 + (t)^(2))^(alpha -(1)/(2))* exp(- x*t), t = 0..infinity)", "(2*((x)/(2))^(alpha))/(sqrt(Pi)*GAMMA(alpha +(1)/(2)))*int((1 + (t)^(2))^(alpha -(1)/(2))* exp(- x*t), t = 0..infinity) = (2*((x)/(2))^(alpha))/(sqrt(Pi)*GAMMA(alpha +(1)/(2)))*int(exp(- x*sinh(tau))*(cosh(tau))^(2*alpha), tau = 0..infinity)" ],
        "freeVariables" : [ "alpha", "x" ],
        "tokenTranslations" : {
          "\\cosh" : "Hyperbolic cosine; Example: \\cosh@@{z}\nWill be translated to: cosh($0)\nRelevant links to definitions:\nDLMF:  http://dlmf.nist.gov/4.28#E2\nMaple: https://www.maplesoft.com/support/help/maple/view.aspx?path=cosh",
          "\\expe" : "Recognizes e with power as the exponential function. It was translated as a function.",
          "\\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",
          "\\sinh" : "Hyperbolic sine; Example: \\sinh@@{z}\nWill be translated to: sinh($0)\nRelevant links to definitions:\nDLMF:  http://dlmf.nist.gov/4.28#E1\nMaple: https://www.maplesoft.com/support/help/maple/view.aspx?path=sinh",
          "\\cpi" : "Pi was translated to: Pi",
          "\\StruveK" : "Associated Struve funtion; Example: \\StruveK{\\nu}@{z}\nWill be translated to: StruveH($0, $1) - BesselY($0, $1)\nRelevant links to definitions:\nDLMF:  http://dlmf.nist.gov/11.2#E5\nMaple: https://www.maplesoft.com/support/help/maple/view.aspx?path=StruveH",
          "\\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)" ],
  "isPartOf" : [ ],
  "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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