FourierCosTransform[Erfc[t], t, x] == (Sqrt[2/Pi] (1/x) Erfi[x/2])/ E^(x^2/4) /; Im[x] == 0

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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> <mrow> <msub> <mi> &#8497;&#119992; </mi> <mi> t </mi> </msub> <mo> [ </mo> <mrow> <mi> erfc </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> <mo> ] </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <msqrt> <mfrac> <mn> 2 </mn> <mi> &#960; </mi> </mfrac> </msqrt> <mo> &#8290; </mo> <mfrac> <mn> 1 </mn> <mi> x </mi> </mfrac> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mo> - </mo> <mfrac> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mn> 4 </mn> </mfrac> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> erfi </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mi> x </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mn> 0 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <times /> <apply> <apply> <ci> Subscript </ci> <ci> &#8497;&#119992; </ci> <ci> t </ci> </apply> <apply> <ci> Erfc </ci> <ci> t </ci> </apply> </apply> <ci> x </ci> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> x </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <ci> Erfi </ci> <apply> <times /> <ci> x </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <eq /> <apply> <imaginary /> <ci> x </ci> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </annotation-xml> </semantics> </math>

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2001-10-29