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]

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</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; 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</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>

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2001-10-29