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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/06.27.21.0092.01









  


  










Input Form





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










Standard Form





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MathML Form







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</apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Erfc </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <sinh /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <cosh /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </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_", "3"], " ", RowBox[List["Cosh", "[", RowBox[List["b_", " ", SuperscriptBox["z_", "2"]]], "]"]], " ", RowBox[List["Erfc", "[", RowBox[List["a_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[FractionBox[RowBox[List["a", " ", "b", " ", SuperscriptBox["z", "3"], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["-", SqrtBox["\[Pi]"]]], "-", RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], "+", "b"]], ")"]], " ", SuperscriptBox["z", "2"]]]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", "b"]], ")"]], " ", SuperscriptBox["z", "2"]]]]]], "+", RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Erf", "[", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", "b"]], ")"]], " ", SuperscriptBox["z", "2"]]]], "]"]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", "b"]], ")"]], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["3", "/", "2"]]]], "+", FractionBox[RowBox[List[SqrtBox["\[Pi]"], "+", RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "+", "b"]], ")"]]]], " ", SuperscriptBox["z", "2"]]]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "+", "b"]], ")"]], " ", SuperscriptBox["z", "2"]]]]]], "-", RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Erf", "[", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "+", "b"]], ")"]], " ", SuperscriptBox["z", "2"]]]], "]"]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "+", "b"]], ")"]], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["3", "/", "2"]]]]]], ")"]]]], RowBox[List["2", " ", SqrtBox["\[Pi]"]]]], "+", FractionBox[RowBox[List[RowBox[List["a", " ", RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], "+", "b"]], ")"]], " ", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "+", "b"]]], " ", RowBox[List["Erf", "[", RowBox[List[SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "+", "b"]]], " ", "z"]], "]"]]]], "+", RowBox[List["a", " ", SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], "+", "b"]]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "+", "b"]], ")"]], " ", RowBox[List["Erfi", "[", RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], "+", "b"]]], " ", "z"]], "]"]]]]]], RowBox[List[SuperscriptBox["a", "4"], "-", SuperscriptBox["b", "2"]]]], "+", RowBox[List["2", " ", RowBox[List["Erfc", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["Cosh", "[", RowBox[List["b", " ", SuperscriptBox["z", "2"]]], "]"]]]], "+", RowBox[List["b", " ", SuperscriptBox["z", "2"], " ", RowBox[List["Sinh", "[", RowBox[List["b", " ", SuperscriptBox["z", "2"]]], "]"]]]]]], ")"]]]]]], RowBox[List["4", " ", SuperscriptBox["b", "2"]]]]]]]]










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





2001-10-29





© 1998-2014 Wolfram Research, Inc.