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ParabolicCylinderD






Mathematica Notation

Traditional Notation









Hypergeometric Functions > ParabolicCylinderD[nu,z] > Integral transforms > Laplace transforms





http://functions.wolfram.com/07.41.22.0001.01









  


  










Input Form





LaplaceTransform[t^(\[Alpha] - 1) ParabolicCylinderD[\[Nu], Sqrt[t]], t, z] == ((2^(\[Nu]/2 - \[Alpha] + 1) Sqrt[Pi] Gamma[2 \[Alpha]])/ Gamma[(1 - \[Nu])/2 + \[Alpha]]) Hypergeometric2F1[\[Alpha], \[Alpha] + 1/2, (1 - \[Nu])/2 + \[Alpha], 1/2 - 2 z] /; (Re[\[Alpha]] > 0 && Re[4 z + 1] > 0) || (0 < Re[\[Alpha]] < -(Re[\[Nu]]/2) && Re[4 z + 1] = 0)










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["LaplaceTransform", "[", RowBox[List[RowBox[List[SuperscriptBox["t", RowBox[List["\[Alpha]", "-", "1"]]], " ", RowBox[List["ParabolicCylinderD", "[", RowBox[List["\[Nu]", ",", SqrtBox["t"]]], "]"]]]], ",", "t", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[FractionBox["\[Nu]", "2"], "-", "\[Alpha]", "+", "1"]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", RowBox[List["2", "\[Alpha]"]], "]"]], " "]], RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], "+", "\[Alpha]"]], "]"]]], RowBox[List["Hypergeometric2F1", "[", RowBox[List["\[Alpha]", ",", RowBox[List["\[Alpha]", "+", FractionBox["1", "2"]]], ",", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], " ", "+", "\[Alpha]"]], ",", RowBox[List[FractionBox["1", "2"], " ", "-", RowBox[List["2", "z"]]]]]], "]"]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Re", "[", "\[Alpha]", "]"]], ">", "0"]], "&&", RowBox[List[RowBox[List["Re", "[", RowBox[List[RowBox[List["4", " ", "z"]], "+", "1"]], "]"]], ">", "0"]]]], ")"]], "||", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["0", "<", RowBox[List["Re", "[", "\[Alpha]", "]"]], "<", RowBox[List["-", FractionBox[RowBox[List["Re", "[", "\[Nu]", "]"]], "2"]]]]], "&&", RowBox[List["Re", "[", RowBox[List[RowBox[List["4", " ", "z"]], "+", "1"]], "]"]]]], "=", "0"]], ")"]]]]]]]]










MathML Form







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</mo> <msqrt> <mi> &#960; </mi> </msqrt> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#945; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#945; </mi> <mo> + </mo> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> &#957; </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> &#945; </mi> <mo> , </mo> <mrow> <mi> &#945; </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> ; </mo> <mrow> <mi> &#945; </mi> <mo> + </mo> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> &#957; </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> ; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; 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</mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> = </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> LaplaceTransform </ci> <apply> <apply> <ci> Subscript </ci> <apply> <ci> ParabolicCylinderD </ci> <apply> <times /> <apply> <power /> <ci> t </ci> <apply> <plus /> <ci> &#945; </ci> <cn type='integer'> -1 </cn> </apply> </apply> <ci> D </ci> </apply> </apply> <ci> &#957; </ci> </apply> <apply> <power /> <ci> t </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <ci> t </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> <apply> <times /> <ci> &#957; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> Gamma </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#945; </ci> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> &#945; </ci> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Hypergeometric2F1 </ci> <ci> &#945; </ci> <apply> <plus /> <ci> &#945; </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <ci> &#945; </ci> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <or /> <apply> <and /> <apply> <gt /> <apply> <real /> <ci> &#945; </ci> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <gt /> <apply> <real /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> z </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <ci> Set </ci> <apply> <and /> <apply> <lt /> <cn type='integer'> 0 </cn> <apply> <real /> <ci> &#945; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <real /> <ci> &#957; </ci> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <real /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> z </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LaplaceTransform", "[", RowBox[List[RowBox[List[SuperscriptBox["t_", RowBox[List["\[Alpha]_", "-", "1"]]], " ", RowBox[List["ParabolicCylinderD", "[", RowBox[List["\[Nu]_", ",", SqrtBox["t_"]]], "]"]]]], ",", "t_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[FractionBox["\[Nu]", "2"], "-", "\[Alpha]", "+", "1"]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", RowBox[List["2", " ", "\[Alpha]"]], "]"]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List["\[Alpha]", ",", RowBox[List["\[Alpha]", "+", FractionBox["1", "2"]]], ",", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], "+", "\[Alpha]"]], ",", RowBox[List[FractionBox["1", "2"], "-", RowBox[List["2", " ", "z"]]]]]], "]"]]]], RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], "+", "\[Alpha]"]], "]"]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Re", "[", "\[Alpha]", "]"]], ">", "0"]], "&&", RowBox[List[RowBox[List["Re", "[", RowBox[List[RowBox[List["4", " ", "z"]], "+", "1"]], "]"]], ">", "0"]]]], ")"]], "||", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["0", "<", RowBox[List["Re", "[", "\[Alpha]", "]"]], "<", RowBox[List["-", FractionBox[RowBox[List["Re", "[", "\[Nu]", "]"]], "2"]]]]], "&&", RowBox[List["Re", "[", RowBox[List[RowBox[List["4", " ", "z"]], "+", "1"]], "]"]]]], "=", "0"]], ")"]]]]]]]]]]










Date Added to functions.wolfram.com (modification date)





2007-05-02





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