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 D_a(z)=\frac{1}{\sqrt{\pi }}{2^{a/2} e^{-\frac{z^2}{4}} \left(\cos \left(\frac{\pi a}{2}\right) \Gamma \left(\frac{a+1}{2}\right) \, _1F_1\left(-\frac{a}{2};\frac{1}{2};\frac{z^2}{2}\right)+\sqrt{2} z \sin \left(\frac{\pi a}{2}\right) \Gamma \left(\frac{a}{2}+1\right) \, _1F_1\left(\frac{1}{2}-\frac{a}{2};\frac{3}{2};\frac{z^2}{2}\right)\right)}.}

${\displaystyle D_{a}(z)={\frac {1}{\sqrt {\pi }}}{2^{a/2}e^{-{\frac {z^{2}}{4}}}\left(\cos \left({\frac {\pi a}{2}}\right)\Gamma \left({\frac {a+1}{2}}\right)\,_{1}F_{1}\left(-{\frac {a}{2}};{\frac {1}{2}};{\frac {z^{2}}{2}}\right)+{\sqrt {2}}z\sin \left({\frac {\pi a}{2}}\right)\Gamma \left({\frac {a}{2}}+1\right)\,_{1}F_{1}\left({\frac {1}{2}}-{\frac {a}{2}};{\frac {3}{2}};{\frac {z^{2}}{2}}\right)\right)}.}$

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

Semantic latex: \WhittakerparaD{a}@{z} = \frac{1}{\sqrt{\cpi}}{2^{a/2} \expe^{-\frac{z^2}{4}}(\cos(\frac{\cpi a}{2}) \Gamma(\frac{a+1}{2})_1 F_1(- \frac{a}{2} ; \frac{1}{2} ; \frac{z^2}{2}) + \sqrt{2} z \sin(\frac{\cpi a}{2}) \Gamma(\frac{a}{2} + 1)_1 F_1(\frac{1}{2} - \frac{a}{2} ; \frac{3}{2} ; \frac{z^2}{2}))}

Confidence: 0.65507224125177

Mathematica

Translation: ParabolicCylinderD[a, z] == Divide[1,Sqrt[Pi]]*(2)^(a/2)* Exp[-Divide[(z)^(2),4]]*(Cos[Divide[Pi*a,2]]*Subscript[\[CapitalGamma][Divide[a + 1,2]], 1]*Subscript[F, 1][-Divide[a,2];Divide[1,2];Divide[(z)^(2),2]]+Sqrt[2]*z*Sin[Divide[Pi*a,2]]*Subscript[\[CapitalGamma][Divide[a,2]+ 1], 1]*Subscript[F, 1][Divide[1,2]-Divide[a,2];Divide[3,2];Divide[(z)^(2),2]])

Information

Sub Equations

ParabolicCylinderD[a, z] = Divide[1,Sqrt[Pi]]*(2)^(a/2)* Exp[-Divide[(z)^(2),4]]*(Cos[Divide[Pi*a,2]]*Subscript[\[CapitalGamma][Divide[a + 1,2]], 1]*Subscript[F, 1][-Divide[a,2];Divide[1,2];Divide[(z)^(2),2]]+Sqrt[2]*z*Sin[Divide[Pi*a,2]]*Subscript[\[CapitalGamma][Divide[a,2]+ 1], 1]*Subscript[F, 1][Divide[1,2]-Divide[a,2];Divide[3,2];Divide[(z)^(2),2]])

Free variables

\[CapitalGamma]

a

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

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

Function without DLMF-Definition. We keep it like it is (but delete prefix \ if necessary).

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

Whittaker's notation D for the parabolic cylinder function; Example: \WhittakerparaD{\nu}@{z}

Will be translated to: ParabolicCylinderD[$0, $1] Relevant links to definitions: DLMF: http://dlmf.nist.gov/12.1 Mathematica: https://reference.wolfram.com/language/ref/ParabolicCylinderD.html

Pi was translated to: Pi

Function without DLMF-Definition. We keep it like it is (but delete prefix \ if necessary).

Tests

Symbolic

Numeric

SymPy

Translation:

Information

Symbol info

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

Tests

Symbolic

Numeric

Maple

Translation: CylinderD(a, z) = (1)/(sqrt(Pi))*(2)^(a/2)* exp(-((z)^(2))/(4))*(cos((Pi*a)/(2))*Gamma((a + 1)/(2))[1]*F[1](-(a)/(2);(1)/(2);((z)^(2))/(2))+sqrt(2)*z*sin((Pi*a)/(2))*Gamma((a)/(2)+ 1)[1]*F[1]((1)/(2)-(a)/(2);(3)/(2);((z)^(2))/(2)))

Information

Sub Equations

CylinderD(a, z) = (1)/(sqrt(Pi))*(2)^(a/2)* exp(-((z)^(2))/(4))*(cos((Pi*a)/(2))*Gamma((a + 1)/(2))[1]*F[1](-(a)/(2);(1)/(2);((z)^(2))/(2))+sqrt(2)*z*sin((Pi*a)/(2))*Gamma((a)/(2)+ 1)[1]*F[1]((1)/(2)-(a)/(2);(3)/(2);((z)^(2))/(2)))

Free variables

Gamma

a

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

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

Function without DLMF-Definition. We keep it like it is (but delete prefix \ if necessary).

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

Whittaker's notation D for the parabolic cylinder function; Example: \WhittakerparaD{\nu}@{z}

Will be translated to: CylinderD($0, $1) Relevant links to definitions: DLMF: http://dlmf.nist.gov/12.1 Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=CylinderD

Pi was translated to: Pi

Function without DLMF-Definition. We keep it like it is (but delete prefix \ if necessary).

Tests

Symbolic

Numeric

Dependency Graph Information

Includes

$a$

$z$

$D_{a}(z)={\frac {1}{\sqrt {\pi }}}{2^{a/2}e^{-{\frac {z^{2}}{4}}}\left(\cos \left({\frac {\pi a}{2}}\right)\Gamma \left({\frac {a+1}{2}}\right)\,_{1}F_{1}\left(-{\frac {a}{2}};{\frac {1}{2}};{\frac {z^{2}}{2}}\right)+{\sqrt {2}}z\sin \left({\frac {\pi a}{2}}\right)\Gamma \left({\frac {a}{2}}+1\right)\,_{1}F_{1}\left({\frac {1}{2}}-{\frac {a}{2}};{\frac {3}{2}};{\frac {z^{2}}{2}}\right)\right)}$

$_{a}$

$D_{p}(x)$

Is part of

$D_{a}(z)={\frac {1}{\sqrt {\pi }}}{2^{a/2}e^{-{\frac {z^{2}}{4}}}\left(\cos \left({\frac {\pi a}{2}}\right)\Gamma \left({\frac {a+1}{2}}\right)\,_{1}F_{1}\left(-{\frac {a}{2}};{\frac {1}{2}};{\frac {z^{2}}{2}}\right)+{\sqrt {2}}z\sin \left({\frac {\pi a}{2}}\right)\Gamma \left({\frac {a}{2}}+1\right)\,_{1}F_{1}\left({\frac {1}{2}}-{\frac {a}{2}};{\frac {3}{2}};{\frac {z^{2}}{2}}\right)\right)}$

Complete translation information:

{
  "id" : "FORMULA_899fac88f5b50fa673182d33f810850c",
  "formula" : "D_a(z)=\\frac{1}{\\sqrt{\\pi }}{2^{a/2} e^{-\\frac{z^2}{4}} \\left(\\cos \\left(\\frac{\\pi  a}{2}\\right) \\Gamma \\left(\\frac{a+1}{2}\\right)  _1F_1\\left(-\\frac{a}{2};\\frac{1}{2};\\frac{z^2}{2}\\right)+\\sqrt{2} z \\sin \\left(\\frac{\\pi  a}{2}\\right) \\Gamma \\left(\\frac{a}{2}+1\\right)  _1F_1\\left(\\frac{1}{2}-\\frac{a}{2};\\frac{3}{2};\\frac{z^2}{2}\\right)\\right)}",
  "semanticFormula" : "\\WhittakerparaD{a}@{z} = \\frac{1}{\\sqrt{\\cpi}}{2^{a/2} \\expe^{-\\frac{z^2}{4}}(\\cos(\\frac{\\cpi a}{2}) \\Gamma(\\frac{a+1}{2})_1 F_1(- \\frac{a}{2} ; \\frac{1}{2} ; \\frac{z^2}{2}) + \\sqrt{2} z \\sin(\\frac{\\cpi a}{2}) \\Gamma(\\frac{a}{2} + 1)_1 F_1(\\frac{1}{2} - \\frac{a}{2} ; \\frac{3}{2} ; \\frac{z^2}{2}))}",
  "confidence" : 0.6550722412517715,
  "translations" : {
    "Mathematica" : {
      "translation" : "ParabolicCylinderD[a, z] == Divide[1,Sqrt[Pi]]*(2)^(a/2)* Exp[-Divide[(z)^(2),4]]*(Cos[Divide[Pi*a,2]]*Subscript[\\[CapitalGamma][Divide[a + 1,2]], 1]*Subscript[F, 1][-Divide[a,2];Divide[1,2];Divide[(z)^(2),2]]+Sqrt[2]*z*Sin[Divide[Pi*a,2]]*Subscript[\\[CapitalGamma][Divide[a,2]+ 1], 1]*Subscript[F, 1][Divide[1,2]-Divide[a,2];Divide[3,2];Divide[(z)^(2),2]])",
      "translationInformation" : {
        "subEquations" : [ "ParabolicCylinderD[a, z] = Divide[1,Sqrt[Pi]]*(2)^(a/2)* Exp[-Divide[(z)^(2),4]]*(Cos[Divide[Pi*a,2]]*Subscript[\\[CapitalGamma][Divide[a + 1,2]], 1]*Subscript[F, 1][-Divide[a,2];Divide[1,2];Divide[(z)^(2),2]]+Sqrt[2]*z*Sin[Divide[Pi*a,2]]*Subscript[\\[CapitalGamma][Divide[a,2]+ 1], 1]*Subscript[F, 1][Divide[1,2]-Divide[a,2];Divide[3,2];Divide[(z)^(2),2]])" ],
        "freeVariables" : [ "\\[CapitalGamma]", "a", "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",
          "\\expe" : "Recognizes e with power as the exponential function. It was translated as a function.",
          "F" : "Function without DLMF-Definition. We keep it like it is (but delete prefix \\ if necessary).",
          "\\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",
          "\\WhittakerparaD" : "Whittaker's notation D for the parabolic cylinder function; Example: \\WhittakerparaD{\\nu}@{z}\nWill be translated to: ParabolicCylinderD[$0, $1]\nRelevant links to definitions:\nDLMF:         http://dlmf.nist.gov/12.1\nMathematica:  https://reference.wolfram.com/language/ref/ParabolicCylinderD.html",
          "\\cpi" : "Pi was translated to: Pi",
          "\\Gamma" : "Function without DLMF-Definition. We keep it like it is (but delete prefix \\ if necessary)."
        }
      },
      "numericResults" : {
        "overallResult" : "SKIPPED",
        "numberOfTests" : 0,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 0,
        "wasAborted" : false,
        "crashed" : false,
        "testCalculationsGroups" : [ ]
      },
      "symbolicResults" : {
        "overallResult" : "SKIPPED",
        "numberOfTests" : 0,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 0,
        "crashed" : false,
        "testCalculationsGroup" : [ ]
      }
    },
    "SymPy" : {
      "translation" : "",
      "translationInformation" : {
        "tokenTranslations" : {
          "Error" : "(LaTeX -> SymPy) No translation possible for given token: Cannot extract information from feature set: \\WhittakerparaD [\\WhittakerparaD]"
        }
      },
      "numericResults" : {
        "overallResult" : "SKIPPED",
        "numberOfTests" : 0,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 0,
        "wasAborted" : false,
        "crashed" : false,
        "testCalculationsGroups" : [ ]
      },
      "symbolicResults" : {
        "overallResult" : "SKIPPED",
        "numberOfTests" : 0,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 0,
        "crashed" : false,
        "testCalculationsGroup" : [ ]
      }
    },
    "Maple" : {
      "translation" : "CylinderD(a, z) = (1)/(sqrt(Pi))*(2)^(a/2)* exp(-((z)^(2))/(4))*(cos((Pi*a)/(2))*Gamma((a + 1)/(2))[1]*F[1](-(a)/(2);(1)/(2);((z)^(2))/(2))+sqrt(2)*z*sin((Pi*a)/(2))*Gamma((a)/(2)+ 1)[1]*F[1]((1)/(2)-(a)/(2);(3)/(2);((z)^(2))/(2)))",
      "translationInformation" : {
        "subEquations" : [ "CylinderD(a, z) = (1)/(sqrt(Pi))*(2)^(a/2)* exp(-((z)^(2))/(4))*(cos((Pi*a)/(2))*Gamma((a + 1)/(2))[1]*F[1](-(a)/(2);(1)/(2);((z)^(2))/(2))+sqrt(2)*z*sin((Pi*a)/(2))*Gamma((a)/(2)+ 1)[1]*F[1]((1)/(2)-(a)/(2);(3)/(2);((z)^(2))/(2)))" ],
        "freeVariables" : [ "Gamma", "a", "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",
          "\\expe" : "Recognizes e with power as the exponential function. It was translated as a function.",
          "F" : "Function without DLMF-Definition. We keep it like it is (but delete prefix \\ if necessary).",
          "\\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",
          "\\WhittakerparaD" : "Whittaker's notation D for the parabolic cylinder function; Example: \\WhittakerparaD{\\nu}@{z}\nWill be translated to: CylinderD($0, $1)\nRelevant links to definitions:\nDLMF:  http://dlmf.nist.gov/12.1\nMaple: https://www.maplesoft.com/support/help/maple/view.aspx?path=CylinderD",
          "\\cpi" : "Pi was translated to: Pi",
          "\\Gamma" : "Function without DLMF-Definition. We keep it like it is (but delete prefix \\ if necessary)."
        }
      },
      "numericResults" : {
        "overallResult" : "SKIPPED",
        "numberOfTests" : 0,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
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        "numberOfErrorTests" : 0,
        "wasAborted" : false,
        "crashed" : false,
        "testCalculationsGroups" : [ ]
      },
      "symbolicResults" : {
        "overallResult" : "SKIPPED",
        "numberOfTests" : 0,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 0,
        "crashed" : false,
        "testCalculationsGroup" : [ ]
      }
    }
  },
  "positions" : [ ],
  "includes" : [ "a", "z", "D_a(z)=\\frac{1}{\\sqrt{\\pi }}{2^{a/2} e^{-\\frac{z^2}{4}} \\left(\\cos \\left(\\frac{\\pi  a}{2}\\right) \\Gamma \\left(\\frac{a+1}{2}\\right) \\, _1F_1\\left(-\\frac{a}{2};\\frac{1}{2};\\frac{z^2}{2}\\right)+\\sqrt{2} z \\sin \\left(\\frac{\\pi  a}{2}\\right) \\Gamma \\left(\\frac{a}{2}+1\\right) \\, _1F_1\\left(\\frac{1}{2}-\\frac{a}{2};\\frac{3}{2};\\frac{z^2}{2}\\right)\\right)}", "_{a}", "D_{p}(x)" ],
  "isPartOf" : [ "D_a(z)=\\frac{1}{\\sqrt{\\pi }}{2^{a/2} e^{-\\frac{z^2}{4}} \\left(\\cos \\left(\\frac{\\pi  a}{2}\\right) \\Gamma \\left(\\frac{a+1}{2}\\right) \\, _1F_1\\left(-\\frac{a}{2};\\frac{1}{2};\\frac{z^2}{2}\\right)+\\sqrt{2} z \\sin \\left(\\frac{\\pi  a}{2}\\right) \\Gamma \\left(\\frac{a}{2}+1\\right) \\, _1F_1\\left(\\frac{1}{2}-\\frac{a}{2};\\frac{3}{2};\\frac{z^2}{2}\\right)\\right)}" ],
  "definiens" : [ ]
}

Specify your own input

Formula input
Wikitext context	[[File:Parabolic cylindrical coordinates.png\|thumb\|right\|350px\|[[Coordinate system#Coordinate surface\|Coordinate surfaces]] of parabolic cylindrical coordinates. Parabolic cylinder functions occur when [[separation of variables]] is used on [[Laplace equation\|Laplace's equation]] in these coordinates]] In [[mathematics]], the '''parabolic cylinder functions''' are [[special function]]s defined as solutions to the differential equation {{NumBlk\|:\|<math>\frac{d^2f}{dz^2} + \left(\tilde{a}z^2+\tilde{b}z+\tilde{c}\right)f=0.</math>\|{{EquationRef\|1}}}} This equation is found when the technique of [[separation of variables]] is used on [[Laplace equation\|Laplace's equation]] when expressed in [[parabolic cylindrical coordinates]]. The above equation may be brought into two distinct forms (A) and (B) by [[completing the square]] and rescaling ''z'', called [[H. F. Weber]]'s equations {{harv\|Weber\|1869}}: :<math>\frac{d^2f}{dz^2} - \left(\tfrac14z^2+a\right)f=0</math>    (A) and :<math>\frac{d^2f}{dz^2} + \left(\tfrac14z^2-a\right)f=0.</math>     (B) If :<math>f(a,z)\,</math> is a solution, then so are :<math>f(a,-z), f(-a,iz)\text{ and }f(-a,-iz).\,</math> If :<math>f(a,z)\,</math> is a solution of equation (A), then :<math>f(-ia,ze^{(1/4)\pi i})\,</math> is a solution of (B), and, by symmetry, :<math>f(-ia,-ze^{(1/4)\pi i}), f(ia,-ze^{-(1/4)\pi i})\text{ and }f(ia,ze^{-(1/4)\pi i})\,</math> are also solutions of (B). ==Solutions== There are independent even and odd solutions of the form (A). These are given by (following the notation of [[Abramowitz and Stegun]] (1965)): :<math>y_1(a;z) = \exp(-z^2/4) \;_1F_1 \left(\tfrac12a+\tfrac14; \; \tfrac12\; ; \; \frac{z^2}{2}\right)\,\,\,\,\,\, (\mathrm{even})</math> and :<math>y_2(a;z) = z\exp(-z^2/4) \;_1F_1 \left(\tfrac12a+\tfrac34; \; \tfrac32\; ; \; \frac{z^2}{2}\right)\,\,\,\,\,\, (\mathrm{odd})</math> where <math>\;_1F_1 (a;b;z)=M(a;b;z)</math> is the [[confluent hypergeometric function]]. Other pairs of independent solutions may be formed from linear combinations of the above solutions (see Abramowitz and Stegun). One such pair is based upon their behavior at infinity: :<math> U(a,z)=\frac{1}{2^\xi\sqrt{\pi}} \left[ \cos(\xi\pi)\Gamma(1/2-\xi)\,y_1(a,z) -\sqrt{2}\sin(\xi\pi)\Gamma(1-\xi)\,y_2(a,z) \right] </math> :<math> V(a,z)=\frac{1}{2^\xi\sqrt{\pi}\Gamma[1/2-a]} \left[ \sin(\xi\pi)\Gamma(1/2-\xi)\,y_1(a,z) +\sqrt{2}\cos(\xi\pi)\Gamma(1-\xi)\,y_2(a,z) \right] </math> where :<math> \xi=\frac{1}{2}a+\frac{1}{4} . </math> The function ''U''(''a'', ''z'') approaches zero for large values of z  and \|arg(''z'')\| < π/2, while ''V''(''a'', ''z'') diverges for large values of positive real ''z'' . :<math> \lim_{z\rightarrow\infty}U(a,z)/e^{-z^2/4}z^{-a-1/2}=1\,\,\,\,(\text{for}\,\|\arg(z)\|<\pi/2) </math> and :<math> \lim_{z\rightarrow\infty}V(a,z)/\sqrt{\frac{2}{\pi}}e^{z^2/4}z^{a-1/2}=1\,\,\,\,(\text{for}\,\arg(z)=0) . </math> For [[half-integer]] values of ''a'', these (that is, ''U'' and ''V'') can be re-expressed in terms of [[Hermite polynomials]]; alternatively, they can also be expressed in terms of [[Bessel function]]s. The functions ''U'' and ''V'' can also be related to the functions ''D<sub>p</sub>''(''x'') (a notation dating back to Whittaker (1902)) that are themselves sometimes called parabolic cylinder functions (see Abramowitz and Stegun (1965)): :<math>U(a,x)=D_{-a-\tfrac12}(x),</math> :<math>V(a,x)=\frac{\Gamma(\tfrac12+a)}{\pi}[\sin( \pi a) D_{-a-\tfrac12}(x)+D_{-a-\tfrac12}(-x)] .</math> Function ''D<sub>a</sub>(z)'' was introduced by Whittaker and Watson as a solution of eq.~({{EquationNote\|1}}) with <math>\tilde a=-\frac14, \tilde b=0, \tilde c=a+\frac12</math> bounded at <math>+\infty</math>. It can be expressed in terms of confluent hypergeometric functions as :<math>D_a(z)=\frac{1}{\sqrt{\pi }}{2^{a/2} e^{-\frac{z^2}{4}} \left(\cos \left(\frac{\pi a}{2}\right) \Gamma \left(\frac{a+1}{2}\right) \, _1F_1\left(-\frac{a}{2};\frac{1}{2};\frac{z^2}{2}\right)+\sqrt{2} z \sin \left(\frac{\pi a}{2}\right) \Gamma \left(\frac{a}{2}+1\right) \, _1F_1\left(\frac{1}{2}-\frac{a}{2};\frac{3}{2};\frac{z^2}{2}\right)\right)}.</math> {{no footnotes\|date=December 2010}} == References == * {{AS ref \|19\|686}} {{springer\|id=W/w097310\|title=Weber equation\|first=N.Kh.\|last= Rozov}} {{dlmf\|id=12\|first=N. M. \|last=Temme}} Weber, H.F. (1869) "Ueber die Integration der partiellen Differentialgleichung <math>\partial^2u/\partial x^2+\partial^2u/\partial y^2+k^2u=0</math>". ''Math. Ann.'', 1, 1–36 Whittaker, E.T. (1902) "On the functions associated with the parabolic cylinder in harmonic analysis" ''Proc. London Math. Soc.''35, 417–427. *Whittaker, E. T. and Watson, G. N. "The Parabolic Cylinder Function." §16.5 in A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, pp. 347-348, 1990. {{DEFAULTSORT:Parabolic Cylinder Function}} [[Category:Special hypergeometric functions]]
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