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{J}_\nu(z)=\cos\frac{\pi\nu}{2}\sum_{k=0}^\infty\frac{(-1)^kz^{2k}}{4^k\Gamma\left(k+\frac{\nu}{2}+1\right)\Gamma\left(k-\frac{\nu}{2}+1\right)}+\sin\frac{\pi\nu}{2}\sum_{k=0}^\infty\frac{(-1)^kz^{2k+1}}{2^{2k+1}\Gamma\left(k+\frac{\nu}{2}+\frac{3}{2}\right)\Gamma\left(k-\frac{\nu}{2}+\frac{3}{2}\right)}}

${\displaystyle \mathbf {J} _{\nu }(z)=\cos {\frac {\pi \nu }{2}}\sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k}}{4^{k}\Gamma \left(k+{\frac {\nu }{2}}+1\right)\Gamma \left(k-{\frac {\nu }{2}}+1\right)}}+\sin {\frac {\pi \nu }{2}}\sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k+1}}{2^{2k+1}\Gamma \left(k+{\frac {\nu }{2}}+{\frac {3}{2}}\right)\Gamma \left(k-{\frac {\nu }{2}}+{\frac {3}{2}}\right)}}}$

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

Semantic latex: \AngerJ{\nu}@{z} = \cos \frac{\cpi \nu}{2} \sum_{k=0}^\infty \frac{(-1)^k z^{2k}}{4^k \Gamma(k + \frac{\nu}{2} + 1) \Gamma(k - \frac{\nu}{2} + 1)} + \sin \frac{\cpi \nu}{2} \sum_{k=0}^\infty \frac{(-1)^k z^{2k+1}}{2^{2k+1} \Gamma(k + \frac{\nu}{2} + \frac{3}{2}) \Gamma(k - \frac{\nu}{2} + \frac{3}{2})}

Confidence: 0.70910057604668

Mathematica

Translation: AngerJ[\[Nu], z] == Cos[Divide[Pi*\[Nu],2]]*Sum[Divide[(- 1)^(k)* (z)^(2*k),(4)^(k)* \[CapitalGamma]*(k +Divide[\[Nu],2]+ 1)*\[CapitalGamma]*(k -Divide[\[Nu],2]+ 1)], {k, 0, Infinity}, GenerateConditions->None]+ Sin[Divide[Pi*\[Nu],2]]*Sum[Divide[(- 1)^(k)* (z)^(2*k + 1),(2)^(2*k + 1)* \[CapitalGamma]*(k +Divide[\[Nu],2]+Divide[3,2])*\[CapitalGamma]*(k -Divide[\[Nu],2]+Divide[3,2])], {k, 0, Infinity}, GenerateConditions->None]

Information

Sub Equations

AngerJ[\[Nu], z] = Cos[Divide[Pi*\[Nu],2]]*Sum[Divide[(- 1)^(k)* (z)^(2*k),(4)^(k)* \[CapitalGamma]*(k +Divide[\[Nu],2]+ 1)*\[CapitalGamma]*(k -Divide[\[Nu],2]+ 1)], {k, 0, Infinity}, GenerateConditions->None]+ Sin[Divide[Pi*\[Nu],2]]*Sum[Divide[(- 1)^(k)* (z)^(2*k + 1),(2)^(2*k + 1)* \[CapitalGamma]*(k +Divide[\[Nu],2]+Divide[3,2])*\[CapitalGamma]*(k -Divide[\[Nu],2]+Divide[3,2])], {k, 0, Infinity}, GenerateConditions->None]

Free variables

\[CapitalGamma]

\[Nu]

z

Symbol info

Cosine; Example: \cos@@{z}

Will be translated to: Cos[$0] Relevant links to definitions: DLMF: http://dlmf.nist.gov/4.14#E2 Mathematica: https://reference.wolfram.com/language/ref/Cos.html

Anger function; Example: \AngerJ{v}@{z}

Will be translated to: AngerJ[$0, $1] Relevant links to definitions: DLMF: http://dlmf.nist.gov/11.10#E1 Mathematica: https://reference.wolfram.com/language/ref/AngerJ.html

Pi was translated to: Pi

Sine; Example: \sin@@{z}

Will be translated to: Sin[$0] Relevant links to definitions: DLMF: http://dlmf.nist.gov/4.14#E1 Mathematica: https://reference.wolfram.com/language/ref/Sin.html

Tests

Symbolic

Test expression: (AngerJ[\[Nu], z])-(Cos[Divide[Pi*\[Nu],2]]*Sum[Divide[(- 1)^(k)* (z)^(2*k),(4)^(k)* \[CapitalGamma]*(k +Divide[\[Nu],2]+ 1)*\[CapitalGamma]*(k -Divide[\[Nu],2]+ 1)], {k, 0, Infinity}, GenerateConditions->None]+ Sin[Divide[Pi*\[Nu],2]]*Sum[Divide[(- 1)^(k)* (z)^(2*k + 1),(2)^(2*k + 1)* \[CapitalGamma]*(k +Divide[\[Nu],2]+Divide[3,2])*\[CapitalGamma]*(k -Divide[\[Nu],2]+Divide[3,2])], {k, 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: \AngerJ [\AngerJ]

Tests

Symbolic

Numeric

Maple

Translation: AngerJ(nu, z) = cos((Pi*nu)/(2))*sum(((- 1)^(k)* (z)^(2*k))/((4)^(k)* Gamma*(k +(nu)/(2)+ 1)*Gamma*(k -(nu)/(2)+ 1)), k = 0..infinity)+ sin((Pi*nu)/(2))*sum(((- 1)^(k)* (z)^(2*k + 1))/((2)^(2*k + 1)* Gamma*(k +(nu)/(2)+(3)/(2))*Gamma*(k -(nu)/(2)+(3)/(2))), k = 0..infinity)

Information

Sub Equations

AngerJ(nu, z) = cos((Pi*nu)/(2))*sum(((- 1)^(k)* (z)^(2*k))/((4)^(k)* Gamma*(k +(nu)/(2)+ 1)*Gamma*(k -(nu)/(2)+ 1)), k = 0..infinity)+ sin((Pi*nu)/(2))*sum(((- 1)^(k)* (z)^(2*k + 1))/((2)^(2*k + 1)* Gamma*(k +(nu)/(2)+(3)/(2))*Gamma*(k -(nu)/(2)+(3)/(2))), k = 0..infinity)

Free variables

Gamma

nu

z

Symbol info

Cosine; Example: \cos@@{z}

Will be translated to: cos($0) Relevant links to definitions: DLMF: http://dlmf.nist.gov/4.14#E2 Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=cos

Anger function; Example: \AngerJ{v}@{z}

Will be translated to: AngerJ($0, $1) Relevant links to definitions: DLMF: http://dlmf.nist.gov/11.10#E1 Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=AngerJ

Pi was translated to: Pi

Sine; Example: \sin@@{z}

Will be translated to: sin($0) Relevant links to definitions: DLMF: http://dlmf.nist.gov/4.14#E1 Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=sin

Tests

Symbolic

Numeric

Dependency Graph Information

Includes

$\mathbf {J} _{\nu }$

$\mathbf {J} _{\nu }(z)=\cos {\frac {\pi \nu }{2}}\sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k}}{4^{k}\Gamma \left(k+{\frac {\nu }{2}}+1\right)\Gamma \left(k-{\frac {\nu }{2}}+1\right)}}+\sin {\frac {\pi \nu }{2}}\sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k+1}}{2^{2k+1}\Gamma \left(k+{\frac {\nu }{2}}+{\frac {3}{2}}\right)\Gamma \left(k-{\frac {\nu }{2}}+{\frac {3}{2}}\right)}}$

$J_{\nu }$

$\nu$

Is part of

$\mathbf {J} _{\nu }(z)=\cos {\frac {\pi \nu }{2}}\sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k}}{4^{k}\Gamma \left(k+{\frac {\nu }{2}}+1\right)\Gamma \left(k-{\frac {\nu }{2}}+1\right)}}+\sin {\frac {\pi \nu }{2}}\sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k+1}}{2^{2k+1}\Gamma \left(k+{\frac {\nu }{2}}+{\frac {3}{2}}\right)\Gamma \left(k-{\frac {\nu }{2}}+{\frac {3}{2}}\right)}}$

Complete translation information:

{
  "id" : "FORMULA_7248b0f569c7cdb22d1430f98ebcf455",
  "formula" : "\\mathbf{J}_\\nu(z)=\\cos\\frac{\\pi\\nu}{2}\\sum_{k=0}^\\infty\\frac{(-1)^k z^{2k}}{4^k\\Gamma\\left(k+\\frac{\\nu}{2}+1\\right)\\Gamma\\left(k-\\frac{\\nu}{2}+1\\right)}+\\sin\\frac{\\pi\\nu}{2}\\sum_{k=0}^\\infty\\frac{(-1)^k z^{2k+1}}{2^{2k+1}\\Gamma\\left(k+\\frac{\\nu}{2}+\\frac{3}{2}\\right)\\Gamma\\left(k-\\frac{\\nu}{2}+\\frac{3}{2}\\right)}",
  "semanticFormula" : "\\AngerJ{\\nu}@{z} = \\cos \\frac{\\cpi \\nu}{2} \\sum_{k=0}^\\infty \\frac{(-1)^k z^{2k}}{4^k \\Gamma(k + \\frac{\\nu}{2} + 1) \\Gamma(k - \\frac{\\nu}{2} + 1)} + \\sin \\frac{\\cpi \\nu}{2} \\sum_{k=0}^\\infty \\frac{(-1)^k z^{2k+1}}{2^{2k+1} \\Gamma(k + \\frac{\\nu}{2} + \\frac{3}{2}) \\Gamma(k - \\frac{\\nu}{2} + \\frac{3}{2})}",
  "confidence" : 0.709100576046682,
  "translations" : {
    "Mathematica" : {
      "translation" : "AngerJ[\\[Nu], z] == Cos[Divide[Pi*\\[Nu],2]]*Sum[Divide[(- 1)^(k)* (z)^(2*k),(4)^(k)* \\[CapitalGamma]*(k +Divide[\\[Nu],2]+ 1)*\\[CapitalGamma]*(k -Divide[\\[Nu],2]+ 1)], {k, 0, Infinity}, GenerateConditions->None]+ Sin[Divide[Pi*\\[Nu],2]]*Sum[Divide[(- 1)^(k)* (z)^(2*k + 1),(2)^(2*k + 1)* \\[CapitalGamma]*(k +Divide[\\[Nu],2]+Divide[3,2])*\\[CapitalGamma]*(k -Divide[\\[Nu],2]+Divide[3,2])], {k, 0, Infinity}, GenerateConditions->None]",
      "translationInformation" : {
        "subEquations" : [ "AngerJ[\\[Nu], z] = Cos[Divide[Pi*\\[Nu],2]]*Sum[Divide[(- 1)^(k)* (z)^(2*k),(4)^(k)* \\[CapitalGamma]*(k +Divide[\\[Nu],2]+ 1)*\\[CapitalGamma]*(k -Divide[\\[Nu],2]+ 1)], {k, 0, Infinity}, GenerateConditions->None]+ Sin[Divide[Pi*\\[Nu],2]]*Sum[Divide[(- 1)^(k)* (z)^(2*k + 1),(2)^(2*k + 1)* \\[CapitalGamma]*(k +Divide[\\[Nu],2]+Divide[3,2])*\\[CapitalGamma]*(k -Divide[\\[Nu],2]+Divide[3,2])], {k, 0, Infinity}, GenerateConditions->None]" ],
        "freeVariables" : [ "\\[CapitalGamma]", "\\[Nu]", "z" ],
        "tokenTranslations" : {
          "\\cos" : "Cosine; Example: \\cos@@{z}\nWill be translated to: Cos[$0]\nRelevant links to definitions:\nDLMF:         http://dlmf.nist.gov/4.14#E2\nMathematica:  https://reference.wolfram.com/language/ref/Cos.html",
          "\\AngerJ" : "Anger function; Example: \\AngerJ{v}@{z}\nWill be translated to: AngerJ[$0, $1]\nRelevant links to definitions:\nDLMF:         http://dlmf.nist.gov/11.10#E1\nMathematica:  https://reference.wolfram.com/language/ref/AngerJ.html",
          "\\cpi" : "Pi was translated to: Pi",
          "\\sin" : "Sine; Example: \\sin@@{z}\nWill be translated to: Sin[$0]\nRelevant links to definitions:\nDLMF:         http://dlmf.nist.gov/4.14#E1\nMathematica:  https://reference.wolfram.com/language/ref/Sin.html"
        }
      },
      "numericResults" : {
        "overallResult" : "SKIPPED",
        "numberOfTests" : 0,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 0,
        "wasAborted" : false,
        "crashed" : false,
        "testCalculationsGroups" : [ ]
      },
      "symbolicResults" : {
        "overallResult" : "ERROR",
        "numberOfTests" : 1,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 1,
        "crashed" : false,
        "testCalculationsGroup" : [ {
          "lhs" : "AngerJ[\\[Nu], z]",
          "rhs" : "Cos[Divide[Pi*\\[Nu],2]]*Sum[Divide[(- 1)^(k)* (z)^(2*k),(4)^(k)* \\[CapitalGamma]*(k +Divide[\\[Nu],2]+ 1)*\\[CapitalGamma]*(k -Divide[\\[Nu],2]+ 1)], {k, 0, Infinity}, GenerateConditions->None]+ Sin[Divide[Pi*\\[Nu],2]]*Sum[Divide[(- 1)^(k)* (z)^(2*k + 1),(2)^(2*k + 1)* \\[CapitalGamma]*(k +Divide[\\[Nu],2]+Divide[3,2])*\\[CapitalGamma]*(k -Divide[\\[Nu],2]+Divide[3,2])], {k, 0, Infinity}, GenerateConditions->None]",
          "testExpression" : "(AngerJ[\\[Nu], z])-(Cos[Divide[Pi*\\[Nu],2]]*Sum[Divide[(- 1)^(k)* (z)^(2*k),(4)^(k)* \\[CapitalGamma]*(k +Divide[\\[Nu],2]+ 1)*\\[CapitalGamma]*(k -Divide[\\[Nu],2]+ 1)], {k, 0, Infinity}, GenerateConditions->None]+ Sin[Divide[Pi*\\[Nu],2]]*Sum[Divide[(- 1)^(k)* (z)^(2*k + 1),(2)^(2*k + 1)* \\[CapitalGamma]*(k +Divide[\\[Nu],2]+Divide[3,2])*\\[CapitalGamma]*(k -Divide[\\[Nu],2]+Divide[3,2])], {k, 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: \\AngerJ [\\AngerJ]"
        }
      }
    },
    "Maple" : {
      "translation" : "AngerJ(nu, z) = cos((Pi*nu)/(2))*sum(((- 1)^(k)* (z)^(2*k))/((4)^(k)* Gamma*(k +(nu)/(2)+ 1)*Gamma*(k -(nu)/(2)+ 1)), k = 0..infinity)+ sin((Pi*nu)/(2))*sum(((- 1)^(k)* (z)^(2*k + 1))/((2)^(2*k + 1)* Gamma*(k +(nu)/(2)+(3)/(2))*Gamma*(k -(nu)/(2)+(3)/(2))), k = 0..infinity)",
      "translationInformation" : {
        "subEquations" : [ "AngerJ(nu, z) = cos((Pi*nu)/(2))*sum(((- 1)^(k)* (z)^(2*k))/((4)^(k)* Gamma*(k +(nu)/(2)+ 1)*Gamma*(k -(nu)/(2)+ 1)), k = 0..infinity)+ sin((Pi*nu)/(2))*sum(((- 1)^(k)* (z)^(2*k + 1))/((2)^(2*k + 1)* Gamma*(k +(nu)/(2)+(3)/(2))*Gamma*(k -(nu)/(2)+(3)/(2))), k = 0..infinity)" ],
        "freeVariables" : [ "Gamma", "nu", "z" ],
        "tokenTranslations" : {
          "\\cos" : "Cosine; Example: \\cos@@{z}\nWill be translated to: cos($0)\nRelevant links to definitions:\nDLMF:  http://dlmf.nist.gov/4.14#E2\nMaple: https://www.maplesoft.com/support/help/maple/view.aspx?path=cos",
          "\\AngerJ" : "Anger function; Example: \\AngerJ{v}@{z}\nWill be translated to: AngerJ($0, $1)\nRelevant links to definitions:\nDLMF:  http://dlmf.nist.gov/11.10#E1\nMaple: https://www.maplesoft.com/support/help/maple/view.aspx?path=AngerJ",
          "\\cpi" : "Pi was translated to: Pi",
          "\\sin" : "Sine; Example: \\sin@@{z}\nWill be translated to: sin($0)\nRelevant links to definitions:\nDLMF:  http://dlmf.nist.gov/4.14#E1\nMaple: https://www.maplesoft.com/support/help/maple/view.aspx?path=sin"
        }
      }
    }
  },
  "positions" : [ ],
  "includes" : [ "\\mathbf{J}_{\\nu}", "\\mathbf{J}_\\nu(z)=\\cos\\frac{\\pi\\nu}{2}\\sum_{k=0}^\\infty\\frac{(-1)^kz^{2k}}{4^k\\Gamma\\left(k+\\frac{\\nu}{2}+1\\right)\\Gamma\\left(k-\\frac{\\nu}{2}+1\\right)}+\\sin\\frac{\\pi\\nu}{2}\\sum_{k=0}^\\infty\\frac{(-1)^kz^{2k+1}}{2^{2k+1}\\Gamma\\left(k+\\frac{\\nu}{2}+\\frac{3}{2}\\right)\\Gamma\\left(k-\\frac{\\nu}{2}+\\frac{3}{2}\\right)}", "J_{\\nu}", "\\nu" ],
  "isPartOf" : [ "\\mathbf{J}_\\nu(z)=\\cos\\frac{\\pi\\nu}{2}\\sum_{k=0}^\\infty\\frac{(-1)^kz^{2k}}{4^k\\Gamma\\left(k+\\frac{\\nu}{2}+1\\right)\\Gamma\\left(k-\\frac{\\nu}{2}+1\\right)}+\\sin\\frac{\\pi\\nu}{2}\\sum_{k=0}^\\infty\\frac{(-1)^kz^{2k+1}}{2^{2k+1}\\Gamma\\left(k+\\frac{\\nu}{2}+\\frac{3}{2}\\right)\\Gamma\\left(k-\\frac{\\nu}{2}+\\frac{3}{2}\\right)}" ],
  "definiens" : [ ]
}

Specify your own input

Formula input
Wikitext context	In mathematics, the '''Anger function''', introduced by {{harvs\|txt\|authorlink=Carl Theodor Anger\|first=C. T.\|last=Anger\|year=1855}}, is a function defined as : <math>\mathbf{J}_\nu(z)=\frac{1}{\pi} \int_0^\pi \cos (\nu\theta-z\sin\theta) \,d\theta</math> and is closely related to [[Bessel function]]s. The '''Weber function''' (also known as '''Lommel-Weber function'''), introduced by {{harvs\|txt\|authorlink=Heinrich Friedrich Weber\|first=H. F.\|last=Weber\|year=1879}}, is a closely related function defined by : <math>\mathbf{E}_\nu(z)=\frac{1}{\pi} \int_0^\pi \sin (\nu\theta-z\sin\theta) \,d\theta</math> and is closely related to [[Bessel function]]s of the second kind. ==Relation between Weber and Anger functions== The Anger and Weber functions are related by :<math> \begin{align} \sin(\pi \nu)\mathbf{J}_\nu(z) &= \cos(\pi\nu)\mathbf{E}_\nu(z)-\mathbf{E}_{-\nu}(z) \\ -\sin(\pi \nu)\mathbf{E}_\nu(z) &= \cos(\pi\nu)\mathbf{J}_\nu(z)-\mathbf{J}_{-\nu}(z) \end{align} </math> so in particular if ν is not an integer they can be expressed as linear combinations of each other. If ν is an integer then Anger functions '''J'''<sub>ν</sub> are the same as Bessel functions ''J''<sub>ν</sub>, and Weber functions can be expressed as finite linear combinations of [[Struve function]]s. ==Power series expansion== The Anger function has the power series expansion<ref name=DLMF>{{dlmf\|id=11.10\|title=Anger-Weber Functions\|first=R. B. \|last=Paris}}</ref> :<math>\mathbf{J}_\nu(z)=\cos\frac{\pi\nu}{2}\sum_{k=0}^\infty\frac{(-1)^kz^{2k}}{4^k\Gamma\left(k+\frac{\nu}{2}+1\right)\Gamma\left(k-\frac{\nu}{2}+1\right)}+\sin\frac{\pi\nu}{2}\sum_{k=0}^\infty\frac{(-1)^kz^{2k+1}}{2^{2k+1}\Gamma\left(k+\frac{\nu}{2}+\frac{3}{2}\right)\Gamma\left(k-\frac{\nu}{2}+\frac{3}{2}\right)}</math> While the Weber function has the power series expansion<ref name=DLMF/> :<math>\mathbf{E}_\nu(z)=\sin\frac{\pi\nu}{2}\sum_{k=0}^\infty\frac{(-1)^kz^{2k}}{4^k\Gamma\left(k+\frac{\nu}{2}+1\right)\Gamma\left(k-\frac{\nu}{2}+1\right)}-\cos\frac{\pi\nu}{2}\sum_{k=0}^\infty\frac{(-1)^kz^{2k+1}}{2^{2k+1}\Gamma\left(k+\frac{\nu}{2}+\frac{3}{2}\right)\Gamma\left(k-\frac{\nu}{2}+\frac{3}{2}\right)}</math> ==Differential equations== The Anger and Weber functions are solutions of inhomogeneous forms of Bessel's equation :<math>z^2y^{\prime\prime} + zy^\prime +(z^2-\nu^2)y = 0 .</math> More precisely, the Anger functions satisfy the equation<ref name=DLMF/> :<math>z^2y^{\prime\prime} + zy^\prime +(z^2-\nu^2)y = \frac{(z-\nu)\sin(\pi \nu)}{\pi} ,</math> and the Weber functions satisfy the equation<ref name=DLMF/> :<math>z^2y^{\prime\prime} + zy^\prime +(z^2-\nu^2)y = -\frac{z+\nu+(z-\nu)\cos(\pi \nu)}{\pi}.</math> ==Recurrence Relations== The Anger function satisfies this inhomogeneous form of [[recurrence relation]]<ref name=DLMF/> :<math>z\mathbf{J}_{\nu-1}(z)+z\mathbf{J}_{\nu+1}(z)=2\nu\mathbf{J}_\nu(z)-\frac{2\sin\pi\nu}{\pi}</math> While the Weber function satisfies this inhomogeneous form of [[recurrence relation]]<ref name=DLMF/> :<math>z\mathbf{E}_{\nu-1}(z)+z\mathbf{E}_{\nu+1}(z)=2\nu\mathbf{E}_\nu(z)-\frac{2(1-\cos\pi\nu)}{\pi}</math> ==Delay Differential Equations== The Anger and Weber functions satisfy these homogeneous forms of [[delay differential equation]]s<ref name=DLMF/> :<math>\mathbf{J}_{\nu-1}(z)-\mathbf{J}_{\nu+1}(z)=2\dfrac{\partial}{\partial z}\mathbf{J}_\nu(z)</math> :<math>\mathbf{E}_{\nu-1}(z)-\mathbf{E}_{\nu+1}(z)=2\dfrac{\partial}{\partial z}\mathbf{E}_\nu(z)</math> The Anger and Weber functions also satisfy these inhomogeneous forms of [[delay differential equation]]s<ref name=DLMF/> :<math>z\dfrac{\partial}{\partial z}\mathbf{J}_\nu(z)\pm\nu\mathbf{J}_\nu(z)=\pm z\mathbf{J}_{\nu\mp1}(z)\pm\frac{\sin\pi\nu}{\pi}</math> :<math>z\dfrac{\partial}{\partial z}\mathbf{E}_\nu(z)\pm\nu\mathbf{E}_\nu(z)=\pm z\mathbf{E}_{\nu\mp1}(z)\pm\frac{1-\cos\pi\nu}{\pi}</math> ==References== {{AS ref\|12\|498}} C.T. Anger, Neueste Schr. d. Naturf. d. Ges. i. Danzig, 5 (1855) pp. 1–29 {{springer\|id=A/a012490\|title=Anger function\|first=A.P.\|last= Prudnikov\|authorlink=Anatolii Platonovich Prudnikov}} {{springer\|id=W/w097320\|title=Weber function\|first=A.P.\|last= Prudnikov}} [[G.N. Watson]], "A treatise on the theory of Bessel functions", 1–2, Cambridge Univ. Press (1952) H.F. Weber, Zurich Vierteljahresschrift, 24 (1879) pp. 33–76 {{Reflist}} [[Category:Special functions]]
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