FourierTransform[Subscript[\[Psi], k][x], x, p] == (Sqrt[Sinh[Pi k]] Gamma[(-(\[DoubleStruckI]/2)) (p + k)] Gamma[(\[DoubleStruckI]/2) (p - k)])/(2^(\[DoubleStruckI] p) (4 Pi))

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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msub> <mi> &#8497; </mi> <mi> x </mi> </msub> <mo> [ </mo> <mrow> <msub> <mi> &#968; </mi> <mi> k </mi> </msub> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> ] </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mi> p </mi> <mo> ) </mo> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <msqrt> <mrow> <mi> sinh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> </msqrt> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mn> 2 </mn> <mrow> <mrow> <mo> - </mo> <mi> i </mi> </mrow> <mo> &#8290; </mo> <mi> p </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> ( </mo> <mtext> </mtext> <mrow> <mrow> <mo> - </mo> <mfrac> <mi> i </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> p </mi> <mo> + </mo> <mtext> </mtext> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mi> i </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> p </mi> <mo> - </mo> <mtext> </mtext> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mtext> </mtext> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> FourierTransform </ci> <apply> <apply> <ci> Subscript </ci> <ci> &#968; </ci> <ci> k </ci> </apply> <ci> x </ci> </apply> <ci> x </ci> <ci> p </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <sinh /> <apply> <times /> <pi /> <ci> k </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> i </ci> </apply> <ci> p </ci> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> i </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <ci> p </ci> <ci> k </ci> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> i </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <ci> p </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2001-10-29