Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail CommentsSign the Guestbook

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

Download All Introductions For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











Erfi






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Erfi[z] > Integration > Indefinite integration > Involving direct function and Gamma-, Beta-, Erf-type functions > Involving erf-type functions and a power function > Involving erfc and power





http://functions.wolfram.com/06.28.21.0138.01









  


  










Input Form





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










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", "2"], " ", RowBox[List["Erfc", "[", RowBox[List["b", " ", "z"]], "]"]], RowBox[List["Erfi", "[", RowBox[List["a", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["3", " ", SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]]]]], RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]], ")"]], " ", SuperscriptBox["z", "2"]]]], " ", SqrtBox["\[Pi]"], " ", "z"]], RowBox[List["a", " ", "b"]]], "+", FractionBox[RowBox[List["\[Pi]", " ", "z"]], RowBox[List["2", " ", "a", " ", "b", " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]], ")"]], " ", SuperscriptBox["z", "2"]]]]]]], "+", FractionBox[RowBox[List["a", " ", "\[Pi]", " ", RowBox[List["Erf", "[", RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]]], " ", "z"]], "]"]]]], RowBox[List[SuperscriptBox["b", "3"], " ", SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]]]]]], "-", FractionBox[RowBox[List["\[Pi]", " ", "z", " ", RowBox[List["Erf", "[", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]], ")"]], " ", SuperscriptBox["z", "2"]]]], "]"]]]], RowBox[List["2", " ", "a", " ", "b", " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]], ")"]], " ", SuperscriptBox["z", "2"]]]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]], " ", "\[Pi]", " ", RowBox[List["Erfc", "[", RowBox[List["b", " ", "z"]], "]"]]]], SuperscriptBox["a", "3"]], "-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]], " ", "\[Pi]", " ", SuperscriptBox["z", "2"], " ", RowBox[List["Erfc", "[", RowBox[List["b", " ", "z"]], "]"]]]], "a"], "-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", SuperscriptBox["b", "2"]]], " ", SuperscriptBox["z", "2"]]]], " ", "\[Pi]", " ", RowBox[List["Erfi", "[", RowBox[List["a", " ", "z"]], "]"]]]], SuperscriptBox["b", "3"]], "-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", SuperscriptBox["b", "2"]]], " ", SuperscriptBox["z", "2"]]]], " ", "\[Pi]", " ", SuperscriptBox["z", "2"], " ", RowBox[List["Erfi", "[", RowBox[List["a", " ", "z"]], "]"]]]], "b"], "+", RowBox[List[SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]], " ", SuperscriptBox["z", "3"], " ", RowBox[List["Erfc", "[", RowBox[List["b", " ", "z"]], "]"]], " ", RowBox[List["Erfi", "[", RowBox[List["a", " ", "z"]], "]"]]]], "+", FractionBox[RowBox[List["b", " ", "\[Pi]", " ", RowBox[List["Erfi", "[", RowBox[List[SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]], " ", "z"]], "]"]]]], RowBox[List[SuperscriptBox["a", "3"], " ", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]]]]]]], ")"]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mo> &#8747; </mo> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <mi> erfc </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> erfi </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> &#960; </mi> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> erfc </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> erfi </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> - </mo> <mfrac> <mrow> <msup> <mi> &#8519; </mi> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msup> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mi> erfc </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> z </mi> <mn> 2 </mn> </msup> </mrow> <mi> a </mi> </mfrac> <mo> - </mo> <mfrac> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> - </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msup> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mi> erfi </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mi> b </mi> </mfrac> <mo> + </mo> <mfrac> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> ( </mo> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msup> <mo> &#8290; </mo> <msqrt> <mi> &#960; </mi> </msqrt> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> </mfrac> <mo> - </mo> <mfrac> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mi> erf </mi> <mo> &#8289; </mo> <mo> ( </mo> <msqrt> <mrow> <mrow> <mo> ( </mo> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> a </mi> <mo> &#8290; </mo> <mi> b </mi> <mo> &#8290; </mo> <msqrt> <mrow> <mrow> <mo> ( </mo> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> </mfrac> <mo> + </mo> <mfrac> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> a </mi> <mo> &#8290; </mo> <mi> b </mi> <mo> &#8290; </mo> <msqrt> <mrow> <mrow> <mo> ( </mo> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> </mfrac> <mo> + </mo> <mfrac> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mi> erf </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msqrt> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <msup> <mi> b </mi> <mn> 3 </mn> </msup> <mo> &#8290; </mo> <msqrt> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> </mfrac> <mo> + </mo> <mfrac> <mrow> <msup> <mi> &#8519; </mi> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msup> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mi> erfc </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <msup> <mi> a </mi> <mn> 3 </mn> </msup> </mfrac> <mo> + </mo> <mfrac> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mi> erfi </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msqrt> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <msup> <mi> a </mi> <mn> 3 </mn> </msup> <mo> &#8290; </mo> <msqrt> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> </mfrac> <mo> - </mo> <mfrac> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> - </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msup> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mi> erfi </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <msup> <mi> b </mi> <mn> 3 </mn> </msup> </mfrac> </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> <ci> Erfc </ci> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> </apply> <apply> <ci> Erfi </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'> 3 </cn> <apply> <power /> <pi /> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <pi /> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <apply> <ci> Erfc </ci> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> </apply> <apply> <ci> Erfi </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <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> <pi /> <apply> <ci> Erfc </ci> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <pi /> <apply> <ci> Erfi </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> </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> <power /> <apply> <times /> <ci> a </ci> <ci> b </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <pi /> <apply> <ci> Erf </ci> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <ci> z </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> <ci> b </ci> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <pi /> <ci> z </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> <ci> b </ci> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <ci> a </ci> <pi /> <apply> <ci> Erf </ci> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <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> <pi /> <apply> <ci> Erfc </ci> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <power /> <ci> a </ci> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <ci> b </ci> <pi /> <apply> <ci> Erfi </ci> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <pi /> <apply> <ci> Erfi </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <power /> <ci> b </ci> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", "2"], " ", RowBox[List["Erfc", "[", RowBox[List["b_", " ", "z_"]], "]"]], " ", RowBox[List["Erfi", "[", RowBox[List["a_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]], ")"]], " ", SuperscriptBox["z", "2"]]]], " ", SqrtBox["\[Pi]"], " ", "z"]], RowBox[List["a", " ", "b"]]], "+", FractionBox[RowBox[List["\[Pi]", " ", "z"]], RowBox[List["2", " ", "a", " ", "b", " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]], ")"]], " ", SuperscriptBox["z", "2"]]]]]]], "+", FractionBox[RowBox[List["a", " ", "\[Pi]", " ", RowBox[List["Erf", "[", RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]]], " ", "z"]], "]"]]]], RowBox[List[SuperscriptBox["b", "3"], " ", SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]]]]]], "-", FractionBox[RowBox[List["\[Pi]", " ", "z", " ", RowBox[List["Erf", "[", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]], ")"]], " ", SuperscriptBox["z", "2"]]]], "]"]]]], RowBox[List["2", " ", "a", " ", "b", " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]], ")"]], " ", SuperscriptBox["z", "2"]]]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]], " ", "\[Pi]", " ", RowBox[List["Erfc", "[", RowBox[List["b", " ", "z"]], "]"]]]], SuperscriptBox["a", "3"]], "-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]], " ", "\[Pi]", " ", SuperscriptBox["z", "2"], " ", RowBox[List["Erfc", "[", RowBox[List["b", " ", "z"]], "]"]]]], "a"], "-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", SuperscriptBox["b", "2"]]], " ", SuperscriptBox["z", "2"]]]], " ", "\[Pi]", " ", RowBox[List["Erfi", "[", RowBox[List["a", " ", "z"]], "]"]]]], SuperscriptBox["b", "3"]], "-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", SuperscriptBox["b", "2"]]], " ", SuperscriptBox["z", "2"]]]], " ", "\[Pi]", " ", SuperscriptBox["z", "2"], " ", RowBox[List["Erfi", "[", RowBox[List["a", " ", "z"]], "]"]]]], "b"], "+", RowBox[List[SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]], " ", SuperscriptBox["z", "3"], " ", RowBox[List["Erfc", "[", RowBox[List["b", " ", "z"]], "]"]], " ", RowBox[List["Erfi", "[", RowBox[List["a", " ", "z"]], "]"]]]], "+", FractionBox[RowBox[List["b", " ", "\[Pi]", " ", RowBox[List["Erfi", "[", RowBox[List[SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]], " ", "z"]], "]"]]]], RowBox[List[SuperscriptBox["a", "3"], " ", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]]]]]]], RowBox[List["3", " ", SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]]]]]]]]]










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





2001-10-29





© 1998-2014 Wolfram Research, Inc.