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Erfc






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Erfc[z] > Integration > Indefinite integration > Involving one direct function and elementary functions > Involving hyperbolic functions and a power function > Involving cosh and power





http://functions.wolfram.com/06.27.21.0087.01









  


  










Input Form





Integrate[z^2 Cosh[b z] Erfc[a z], z] == (1/(8 a^4 b^3 Sqrt[Pi])) ((8 a^3 b - 2 a b^3 + 8 a^3 b E^(2 b z) - 2 a b^3 E^(2 b z) + 4 a^3 b^2 z - 4 a^3 b^2 E^(2 b z) z - (8 a^4 - 2 a^2 b^2 + b^4) E^((b + 2 a^2 z)^2/(4 a^2)) Sqrt[Pi] Erf[b/(2 a) - a z] - (8 a^4 - 2 a^2 b^2 + b^4) E^((b + 2 a^2 z)^2/(4 a^2)) Sqrt[Pi] Erf[b/(2 a) + a z] - 8 a^4 E^(a^2 z^2) Sqrt[Pi] Erfc[a z] + 8 a^4 E^(z (2 b + a^2 z)) Sqrt[Pi] Erfc[a z] - 8 a^4 b E^(a^2 z^2) Sqrt[Pi] z Erfc[a z] - 8 a^4 b E^(z (2 b + a^2 z)) Sqrt[Pi] z Erfc[a z] + 8 a^4 b^2 E^(z (b + a^2 z)) Sqrt[Pi] z^2 Erfc[a z] Sinh[b z])/E^(z (b + a^2 z)))










Standard Form





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MathML Form







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<mi> &#960; </mi> </msqrt> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <mi> erfc </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sinh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <mi> b </mi> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> b </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <mi> b </mi> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 3 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> b </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 3 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> b </mi> <mn> 3 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> b </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> <mo> &#8290; </mo> <mi> a </mi> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> b </mi> <mn> 3 </mn> </msup> <mo> &#8290; </mo> <mi> a </mi> </mrow> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <msup> <mi> b </mi> <mn> 4 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> </msup> <mo> &#8290; </mo> <msqrt> <mi> &#960; </mi> </msqrt> <mo> &#8290; </mo> <mrow> <mi> erf </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mi> b </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> a </mi> </mrow> </mfrac> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <msup> <mi> b </mi> <mn> 4 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> </msup> <mo> &#8290; </mo> <msqrt> <mi> &#960; </mi> </msqrt> <mo> &#8290; </mo> <mrow> <mi> erf </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mi> b </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> a </mi> </mrow> </mfrac> <mo> - </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <cosh /> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> </apply> <apply> <ci> Erfc </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <ci> z </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <ci> b </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -8 </cn> <ci> b </ci> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> <apply> <ci> Erfc </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 8 </cn> <ci> b </ci> <apply> <power /> <exponentiale /> <apply> <times /> <ci> z </ci> <apply> <plus /> <apply> <times /> <ci> z </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> </apply> </apply> </apply> </apply> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> <apply> <ci> Erfc </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> Erfc </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <ci> z </ci> <apply> <plus /> <apply> <times /> <ci> z </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> </apply> </apply> </apply> </apply> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> Erfc </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <ci> z </ci> <apply> <plus /> <apply> <times /> <ci> z </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <ci> b </ci> </apply> </apply> </apply> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Erfc </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <sinh /> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 8 </cn> <ci> b </ci> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 8 </cn> <ci> b </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> <ci> z </ci> </apply> </apply> <ci> z </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <ci> z </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> <ci> z </ci> </apply> </apply> <ci> a </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 3 </cn> </apply> <ci> a </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <ci> b </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> Erf </ci> <apply> <plus /> <apply> <times /> <ci> b </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <ci> b </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> Erf </ci> <apply> <plus /> <apply> <times /> <ci> b </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> </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