Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail CommentsSign the Guestbook

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











Exp






Mathematica Notation

Traditional Notation









Elementary Functions > Exp[z] > Equations involving this function > Differential equations > Schroedinger equations





http://functions.wolfram.com/01.03.15.0005.01









  


  










Input Form





WignerTransform[Subscript[\[Psi], k][y], y, {x, p}] == (1/4) I Pi Sinh[Pi k] ((E^(2 I (k - p) x) cM HypergeometricPFQRegularized[ {}, {1 + I k, 1 - I p, 1 + I (k - p)}, E^(4 x)/16])/4^(I (k - p)) - (4^(I (k - p)) cM HypergeometricPFQRegularized[{}, {1 - I k, 1 + I p, 1 - I (k - p)}, E^(4 x)/16])/E^(2 I (k - p) x) + (4^(I (k + p)) cP HypergeometricPFQRegularized[{}, {1 - I k, 1 - I p, 1 - I (k + p)}, E^(4 x)/16])/E^(2 I (k + p) x) - (E^(2 I (k + p) x) cP HypergeometricPFQRegularized[{}, {1 + I k, 1 + I p, 1 + I (k + p)}, E^(4 x)/16])/4^(I (k + p))) /; cP == Csch[k Pi] Csch[(k + p) Pi] Csch[p Pi] && cM == Csch[k Pi] Csch[(k - p) Pi] Csch[p Pi]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["WignerTransform", "[", RowBox[List[RowBox[List[SubscriptBox["\[Psi]", "k"], "[", "y", "]"]], ",", "y", ",", RowBox[List["{", RowBox[List["x", ",", "p"]], "}"]]]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", "4"], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["Sinh", "[", RowBox[List["\[Pi]", " ", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["4", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List["k", "-", "p"]], ")"]]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["k", "-", "p"]], ")"]], " ", "x"]]], " ", "cM", " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", "k"]]]], ",", RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", "p"]]]], ",", RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["k", "-", "p"]], ")"]]]]]]]], "}"]], ",", FractionBox[SuperscriptBox["\[ExponentialE]", RowBox[List["4", " ", "x"]]], "16"]]], "]"]]]], "-", RowBox[List[SuperscriptBox["4", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["k", "-", "p"]], ")"]]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["k", "-", "p"]], ")"]], " ", "x"]]], " ", "cM", " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", "k"]]]], ",", RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", "p"]]]], ",", RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["k", "-", "p"]], ")"]]]]]]]], "}"]], ",", FractionBox[SuperscriptBox["\[ExponentialE]", RowBox[List["4", " ", "x"]]], "16"]]], "]"]]]], "+", RowBox[List[SuperscriptBox["4", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["k", "+", "p"]], ")"]]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["k", "+", "p"]], ")"]], " ", "x"]]], " ", "cP", " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", "k"]]]], ",", RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", "p"]]]], ",", RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["k", "+", "p"]], ")"]]]]]]]], "}"]], ",", FractionBox[SuperscriptBox["\[ExponentialE]", RowBox[List["4", " ", "x"]]], "16"]]], "]"]]]], "-", RowBox[List[SuperscriptBox["4", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List["k", "+", "p"]], ")"]]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["k", "+", "p"]], ")"]], " ", "x"]]], " ", "cP", " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", "k"]]]], ",", RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", "p"]]]], ",", RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["k", "+", "p"]], ")"]]]]]]]], "}"]], ",", FractionBox[SuperscriptBox["\[ExponentialE]", RowBox[List["4", " ", "x"]]], "16"]]], "]"]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["cP", "\[Equal]", RowBox[List[RowBox[List["Csch", "[", RowBox[List["k", " ", "\[Pi]"]], "]"]], " ", RowBox[List["Csch", "[", RowBox[List[RowBox[List["(", RowBox[List["k", "+", "p"]], ")"]], " ", "\[Pi]"]], "]"]], " ", RowBox[List["Csch", "[", RowBox[List["p", " ", "\[Pi]"]], "]"]]]]]], "\[And]", RowBox[List["cM", "\[Equal]", RowBox[List[RowBox[List["Csch", "[", RowBox[List["k", " ", "\[Pi]"]], "]"]], " ", RowBox[List["Csch", "[", RowBox[List[RowBox[List["(", RowBox[List["k", "-", "p"]], ")"]], " ", "\[Pi]"]], "]"]], " ", RowBox[List["Csch", "[", RowBox[List["p", " ", "\[Pi]"]], "]"]]]]]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mrow> <msub> <mi> &#119986; </mi> <mi> y </mi> </msub> <mo> [ </mo> <mrow> <msub> <mi> &#968; </mi> <mi> k </mi> </msub> <mo> ( </mo> <mi> y </mi> <mo> ) </mo> </mrow> <mo> ] </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> x </mi> <mo> , </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#10869; </mo> <mtext> </mtext> <mrow> <munderover> <mo> &#8747; </mo> <mrow> <mo> - </mo> <mi> &#8734; </mi> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <msup> <mi> e </mi> <mrow> <mrow> <mo> - </mo> <mi> i </mi> </mrow> <mo> &#8290; </mo> <mi> y </mi> <mo> &#8290; </mo> <mi> p </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <msub> <mstyle maxsize='0.6'> <mover> <mi> &#968; </mi> <mo> _ </mo> </mover> </mstyle> <mi> k </mi> </msub> <mo> ( </mo> <mrow> <mi> x </mi> <mo> - </mo> <mfrac> <mi> y </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> &#968; </mi> <mi> k </mi> </msub> <mo> ( </mo> <mrow> <mi> x </mi> <mo> + </mo> <mfrac> <mi> y </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> d </mi> <mo> &#8290; </mo> <mtext> &#8203; </mtext> <mi> y </mi> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mi> i </mi> <mn> 4 </mn> </mfrac> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <msqrt> <mrow> <mi> sinh </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mtext> &#62371; </mtext> <mtext> </mtext> <mrow> <mrow> <msub> <mi> c </mi> <mo> - </mo> </msub> <mo> &#8290; </mo> <msup> <mn> 4 </mn> <mrow> <mrow> <mo> - </mo> <mi> i </mi> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> &#8290; </mo> <msub> <mrow> <msup> <mi> e </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> i </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> x </mi> </mrow> </msup> <mtext> </mtext> </mrow> <mrow> <mn> 0 </mn> <mtext> &#62338; </mtext> </mrow> </msub> <mo> &#8290; </mo> <mrow> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> i </mi> <mo> &#8290; </mo> <mi> k </mi> </mrow> </mrow> <mo> , </mo> <mtext> </mtext> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> i </mi> <mo> &#8290; </mo> <mi> p </mi> </mrow> </mrow> <mo> , </mo> <mtext> </mtext> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> i </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <mrow> <msup> <mi> e </mi> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> x </mi> </mrow> </msup> <mo> / </mo> <mn> 16 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mtext> &#62371; </mtext> <mtext> </mtext> <mrow> <msub> <mi> c </mi> <mo> + </mo> </msub> <mo> &#8290; </mo> <msup> <mn> 4 </mn> <mrow> <mi> i </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> &#8290; </mo> <msub> <mrow> <msup> <mi> e </mi> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <mi> i </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> x </mi> </mrow> </msup> <mtext> </mtext> </mrow> <mrow> <mn> 0 </mn> <mtext> &#62338; </mtext> </mrow> </msub> <mo> &#8290; </mo> <mrow> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> i </mi> <mo> &#8290; </mo> <mi> k </mi> </mrow> </mrow> <mo> , </mo> <mtext> </mtext> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> i </mi> <mo> &#8290; </mo> <mi> p </mi> </mrow> </mrow> <mo> , </mo> <mtext> </mtext> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> i </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <mrow> <msup> <mi> e </mi> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> x </mi> </mrow> </msup> <mo> / </mo> <mn> 16 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mtext> &#62371; </mtext> <mtext> </mtext> <mrow> <msub> <mi> c </mi> <mo> + </mo> </msub> <mo> &#8290; </mo> <msup> <mn> 4 </mn> <mrow> <mrow> <mo> - </mo> <mi> i </mi> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> &#8290; </mo> <msub> <mrow> <msup> <mi> e </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> i </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> x </mi> </mrow> </msup> <mtext> </mtext> </mrow> <mrow> <mn> 0 </mn> <mtext> &#62338; </mtext> </mrow> </msub> <mo> &#8290; </mo> <mrow> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> i </mi> <mo> &#8290; </mo> <mi> k </mi> </mrow> </mrow> <mo> , </mo> <mtext> </mtext> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> i </mi> <mo> &#8290; </mo> <mi> p </mi> </mrow> </mrow> <mo> , </mo> <mtext> </mtext> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> i </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <mrow> <msup> <mi> e </mi> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> x </mi> </mrow> </msup> <mo> / </mo> <mn> 16 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mtext> &#62371; </mtext> <mtext> </mtext> <mrow> <msub> <mi> c </mi> <mo> - </mo> </msub> <mo> &#8290; </mo> <msup> <mn> 4 </mn> <mrow> <mi> i </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> &#8290; </mo> <msub> <mrow> <msup> <mi> e </mi> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <mi> i </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> x </mi> </mrow> </msup> <mtext> </mtext> </mrow> <mrow> <mn> 0 </mn> <mtext> &#62338; </mtext> </mrow> </msub> <mo> &#8290; </mo> <mrow> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> i </mi> <mo> &#8290; </mo> <mi> k </mi> </mrow> </mrow> <mo> , </mo> <mtext> </mtext> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> i </mi> <mo> &#8290; </mo> <mi> p </mi> </mrow> </mrow> <mo> , </mo> <mtext> </mtext> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> i </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <mrow> <msup> <mi> e </mi> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> x </mi> </mrow> </msup> <mo> / </mo> <mn> 16 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mtext> &#62371; </mtext> <mrow> <msub> <mi> c </mi> <mo> &#177; </mo> </msub> <mo> &#10869; </mo> <mrow> <mrow> <mi> csch </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> csch </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> p </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mrow> <mi> csch </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> &#177; </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> <mo> . </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <mrow> <mrow> <msub> <mi> &#119986; </mi> <mi> y </mi> </msub> <mo> [ </mo> <mrow> <msub> <mi> &#968; </mi> <mi> k </mi> </msub> <mo> ( </mo> <mi> y </mi> <mo> ) </mo> </mrow> <mo> ] </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> x </mi> <mo> , </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#10869; </mo> <mtext> </mtext> <mrow> <munderover> <mo> &#8747; </mo> <mrow> <mo> - </mo> <mi> &#8734; </mi> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <msup> <mi> e </mi> <mrow> <mrow> <mo> - </mo> <mi> i </mi> </mrow> <mo> &#8290; </mo> <mi> y </mi> <mo> &#8290; </mo> <mi> p </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <msub> <mstyle maxsize='0.6'> <mover> <mi> &#968; </mi> <mo> _ </mo> </mover> </mstyle> <mi> k </mi> </msub> <mo> ( </mo> <mrow> <mi> x </mi> <mo> - </mo> <mfrac> <mi> y </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> &#968; </mi> <mi> k </mi> </msub> <mo> ( </mo> <mrow> <mi> x </mi> <mo> + </mo> <mfrac> <mi> y </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> d </mi> <mo> &#8290; </mo> <mtext> &#8203; </mtext> <mi> y </mi> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mi> i </mi> <mn> 4 </mn> </mfrac> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <msqrt> <mrow> <mi> sinh </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mtext> &#62371; </mtext> <mtext> </mtext> <mrow> <mrow> <msub> <mi> c </mi> <mo> - </mo> </msub> <mo> &#8290; </mo> <msup> <mn> 4 </mn> <mrow> <mrow> <mo> - </mo> <mi> i </mi> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> &#8290; </mo> <msub> <mrow> <msup> <mi> e </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> i </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> x </mi> </mrow> </msup> <mtext> </mtext> </mrow> <mrow> <mn> 0 </mn> <mtext> &#62338; </mtext> </mrow> </msub> <mo> &#8290; </mo> <mrow> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> i </mi> <mo> &#8290; </mo> <mi> k </mi> </mrow> </mrow> <mo> , </mo> <mtext> </mtext> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> i </mi> <mo> &#8290; </mo> <mi> p </mi> </mrow> </mrow> <mo> , </mo> <mtext> </mtext> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> i </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <mrow> <msup> <mi> e </mi> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> x </mi> </mrow> </msup> <mo> / </mo> <mn> 16 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mtext> &#62371; </mtext> <mtext> </mtext> <mrow> <msub> <mi> c </mi> <mo> + </mo> </msub> <mo> &#8290; </mo> <msup> <mn> 4 </mn> <mrow> <mi> i </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> &#8290; </mo> <msub> <mrow> <msup> <mi> e </mi> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <mi> i </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> x </mi> </mrow> </msup> <mtext> </mtext> </mrow> <mrow> <mn> 0 </mn> <mtext> &#62338; </mtext> </mrow> </msub> <mo> &#8290; </mo> <mrow> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> i </mi> <mo> &#8290; </mo> <mi> k </mi> </mrow> </mrow> <mo> , </mo> <mtext> </mtext> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> i </mi> <mo> &#8290; </mo> <mi> p </mi> </mrow> </mrow> <mo> , </mo> <mtext> </mtext> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> i </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <mrow> <msup> <mi> e </mi> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> x </mi> </mrow> </msup> <mo> / </mo> <mn> 16 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mtext> &#62371; </mtext> <mtext> </mtext> <mrow> <msub> <mi> c </mi> <mo> + </mo> </msub> <mo> &#8290; </mo> <msup> <mn> 4 </mn> <mrow> <mrow> <mo> - </mo> <mi> i </mi> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> &#8290; </mo> <msub> <mrow> <msup> <mi> e </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> i </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> x </mi> </mrow> </msup> <mtext> </mtext> </mrow> <mrow> <mn> 0 </mn> <mtext> &#62338; </mtext> </mrow> </msub> <mo> &#8290; </mo> <mrow> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> i </mi> <mo> &#8290; </mo> <mi> k </mi> </mrow> </mrow> <mo> , </mo> <mtext> </mtext> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> i </mi> <mo> &#8290; </mo> <mi> p </mi> </mrow> </mrow> <mo> , </mo> <mtext> </mtext> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> i </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <mrow> <msup> <mi> e </mi> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> x </mi> </mrow> </msup> <mo> / </mo> <mn> 16 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mtext> &#62371; </mtext> <mtext> </mtext> <mrow> <msub> <mi> c </mi> <mo> - </mo> </msub> <mo> &#8290; </mo> <msup> <mn> 4 </mn> <mrow> <mi> i </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> &#8290; </mo> <msub> <mrow> <msup> <mi> e </mi> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <mi> i </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> x </mi> </mrow> </msup> <mtext> </mtext> </mrow> <mrow> <mn> 0 </mn> <mtext> &#62338; </mtext> </mrow> </msub> <mo> &#8290; </mo> <mrow> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> i </mi> <mo> &#8290; </mo> <mi> k </mi> </mrow> </mrow> <mo> , </mo> <mtext> </mtext> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> i </mi> <mo> &#8290; </mo> <mi> p </mi> </mrow> </mrow> <mo> , </mo> <mtext> </mtext> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> i </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <mrow> <msup> <mi> e </mi> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> x </mi> </mrow> </msup> <mo> / </mo> <mn> 16 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mtext> &#62371; </mtext> <mrow> <msub> <mi> c </mi> <mo> &#177; </mo> </msub> <mo> &#10869; </mo> <mrow> <mrow> <mi> csch </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> csch </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> p </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mrow> <mi> csch </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> &#177; </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> <mo> . </mo> </mrow> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["WignerTransform", "[", RowBox[List[RowBox[List[SubscriptBox["\[Psi]", "k"], "[", "y_", "]"]], ",", "y_", ",", RowBox[List["{", RowBox[List["x_", ",", "p_"]], "}"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["Sinh", "[", RowBox[List["\[Pi]", " ", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["4", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List["k", "-", "p"]], ")"]]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["k", "-", "p"]], ")"]], " ", "x"]]], " ", "cM", " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", "k"]]]], ",", RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", "p"]]]], ",", RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["k", "-", "p"]], ")"]]]]]]]], "}"]], ",", FractionBox[SuperscriptBox["\[ExponentialE]", RowBox[List["4", " ", "x"]]], "16"]]], "]"]]]], "-", RowBox[List[SuperscriptBox["4", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["k", "-", "p"]], ")"]]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["k", "-", "p"]], ")"]], " ", "x"]]], " ", "cM", " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", "k"]]]], ",", RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", "p"]]]], ",", RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["k", "-", "p"]], ")"]]]]]]]], "}"]], ",", FractionBox[SuperscriptBox["\[ExponentialE]", RowBox[List["4", " ", "x"]]], "16"]]], "]"]]]], "+", RowBox[List[SuperscriptBox["4", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["k", "+", "p"]], ")"]]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["k", "+", "p"]], ")"]], " ", "x"]]], " ", "cP", " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", "k"]]]], ",", RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", "p"]]]], ",", RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["k", "+", "p"]], ")"]]]]]]]], "}"]], ",", FractionBox[SuperscriptBox["\[ExponentialE]", RowBox[List["4", " ", "x"]]], "16"]]], "]"]]]], "-", RowBox[List[SuperscriptBox["4", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List["k", "+", "p"]], ")"]]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["k", "+", "p"]], ")"]], " ", "x"]]], " ", "cP", " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", "k"]]]], ",", RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", "p"]]]], ",", RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["k", "+", "p"]], ")"]]]]]]]], "}"]], ",", FractionBox[SuperscriptBox["\[ExponentialE]", RowBox[List["4", " ", "x"]]], "16"]]], "]"]]]]]], ")"]]]], "/;", RowBox[List[RowBox[List["cP", "\[Equal]", RowBox[List[RowBox[List["Csch", "[", RowBox[List["k", " ", "\[Pi]"]], "]"]], " ", RowBox[List["Csch", "[", RowBox[List[RowBox[List["(", RowBox[List["k", "+", "p"]], ")"]], " ", "\[Pi]"]], "]"]], " ", RowBox[List["Csch", "[", RowBox[List["p", " ", "\[Pi]"]], "]"]]]]]], "&&", RowBox[List["cM", "\[Equal]", RowBox[List[RowBox[List["Csch", "[", RowBox[List["k", " ", "\[Pi]"]], "]"]], " ", RowBox[List["Csch", "[", RowBox[List[RowBox[List["(", RowBox[List["k", "-", "p"]], ")"]], " ", "\[Pi]"]], "]"]], " ", RowBox[List["Csch", "[", RowBox[List["p", " ", "\[Pi]"]], "]"]]]]]]]]]]]]]]










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





2001-10-29





© 1998- Wolfram Research, Inc.