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Erfc






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Erfc[z] > Integration > Indefinite integration > Involving functions of the direct function and elementary functions > Involving elementary functions of the direct function and elementary functions > Involving products of the direct function and a power function





http://functions.wolfram.com/06.27.21.0125.01









  


  










Input Form





Integrate[z^(\[Alpha] - 1) Erfc[a z] Erfc[b z], z] == (z^\[Alpha] Erfc[a z] Erfc[b z])/\[Alpha] - (a z^(1 + \[Alpha]) (a^2 z^2)^(-(1/2) - \[Alpha]/2) Gamma[(1 + \[Alpha])/2, a^2 z^2])/(Sqrt[Pi] \[Alpha]) - (b z^(1 + \[Alpha]) (b^2 z^2)^(-(1/2) - \[Alpha]/2) Gamma[(1 + \[Alpha])/2, b^2 z^2])/(Sqrt[Pi] \[Alpha]) + ((1/(a Pi \[Alpha])) 2 b z^\[Alpha] Sum[((-1)^k b^(2 k) Gamma[\[Alpha]/2 + k + 1, a^2 z^2])/ (a^(2 k) ((1 + 2 k) k!)), {k, 0, Infinity}])/(a^2 z^2)^(\[Alpha]/2) + ((1/(b Pi \[Alpha])) 2 a z^\[Alpha] Sum[((-1)^k a^(2 k) Gamma[\[Alpha]/2 + k + 1, b^2 z^2])/ (b^(2 k) ((1 + 2 k) k!)), {k, 0, Infinity}])/(b^2 z^2)^(\[Alpha]/2)










Standard Form





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MathML Form







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</ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <factorial /> <ci> k </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> z </ci> <apply> <plus /> <ci> &#945; </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> &#945; </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> &#945; </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <times /> <apply> <plus /> <ci> &#945; </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2001-10-29





© 1998- Wolfram Research, Inc.