html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 Erfi

 http://functions.wolfram.com/06.28.21.0139.01

 Input Form

 Integrate[z^(\[Alpha] - 1) Erfc[a z] Erfi[a z], z] == (z^\[Alpha]/(2 \[Alpha])) (2 Erfi[a z] + ((2 a z)/(((-a^2) z^2)^((\[Alpha] + 1)/2) Sqrt[Pi])) Gamma[(1 + \[Alpha])/2, (-a^2) z^2] - ((8 a^2 z^2)/Pi) HypergeometricPFQ[{1/2, 1}, {3/4, 5/4, 3/2}, (a^4 z^4)/4] + Sqrt[2] a^2 z^2 Gamma[(2 + \[Alpha])/4] HypergeometricPFQRegularized[ {1, (2 + \[Alpha])/4}, {3/4, 5/4, (6 + \[Alpha])/4}, (a^4 z^4)/4])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", RowBox[List["\[Alpha]", "-", "1"]]], " ", RowBox[List["Erfc", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["Erfi", "[", RowBox[List["a", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox[SuperscriptBox["z", "\[Alpha]"], RowBox[List["2", " ", "\[Alpha]"]]], RowBox[List["(", RowBox[List[RowBox[List["2", RowBox[List["Erfi", "[", RowBox[List["a", " ", "z"]], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List["2", "a", " ", "z", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["-", FractionBox[RowBox[List["\[Alpha]", "+", "1"]], "2"]]]]]], SqrtBox["\[Pi]"]], RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "\[Alpha]"]], "2"], ",", RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], " ", SuperscriptBox["z", "2"]]]]], "]"]]]], "-", RowBox[List[FractionBox[RowBox[List["8", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]], "\[Pi]"], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "4"], ",", FractionBox["5", "4"], ",", FractionBox["3", "2"]]], "}"]], ",", FractionBox[RowBox[List[SuperscriptBox["a", "4"], " ", SuperscriptBox["z", "4"]]], "4"]]], "]"]]]], "+", RowBox[List[SqrtBox["2"], " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"], " ", RowBox[List["Gamma", "[", FractionBox[RowBox[List["2", "+", "\[Alpha]"]], "4"], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", FractionBox[RowBox[List["2", "+", "\[Alpha]"]], "4"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "4"], ",", FractionBox["5", "4"], ",", FractionBox[RowBox[List["6", "+", "\[Alpha]"]], "4"]]], "}"]], ",", FractionBox[RowBox[List[SuperscriptBox["a", "4"], " ", SuperscriptBox["z", "4"]]], "4"]]], "]"]]]]]], ")"]]]]]]]]

 MathML Form

 z α - 1 erfc ( a z ) erfi ( a z ) z z α 2 α ( 2 erfi ( a z ) + 2 a z ( - a 2 z 2 ) - α + 1 2 π Γ ( α + 1 2 , - a 2 z 2 ) - 8 a 2 z 2 π 2 F 3 ( 1 2 , 1 ; 3 4 , 5 4 , 3 2 ; a 4 z 4 4 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["3", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox["1", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["1", HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox["3", "4"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox["5", "4"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox["3", "2"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[FractionBox[RowBox[List[SuperscriptBox["a", "4"], " ", SuperscriptBox["z", "4"]]], "4"], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] + 2 a 2 z 2 Γ ( α + 2 4 ) 2 F ~ 3 ( 1 , α + 2 4 ; 3 4 , 5 4 , α + 6 4 ; a 4 z 4 4 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["3", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["\[Alpha]", "+", "2"]], "4"], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox["3", "4"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[FractionBox["5", "4"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["\[Alpha]", "+", "6"]], "4"], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[FractionBox[RowBox[List[SuperscriptBox["a", "4"], " ", SuperscriptBox["z", "4"]]], "4"], HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] ) z z α -1 Erfc a z Erfi a z z α 2 α -1 2 Erfi a z 2 a z -1 a 2 z 2 -1 α 1 2 -1 1 2 -1 Gamma α 1 2 -1 -1 a 2 z 2 -1 8 a 2 z 2 -1 HypergeometricPFQ 1 2 1 3 4 5 4 3 2 a 4 z 4 4 -1 2 1 2 a 2 z 2 Gamma α 2 4 -1 HypergeometricPFQRegularized 1 α 2 4 -1 3 4 5 4 α 6 4 -1 a 4 z 4 4 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", RowBox[List["\[Alpha]_", "-", "1"]]], " ", RowBox[List["Erfc", "[", RowBox[List["a_", " ", "z_"]], "]"]], " ", RowBox[List["Erfi", "[", RowBox[List["a_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox["z", "\[Alpha]"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["Erfi", "[", RowBox[List["a", " ", "z"]], "]"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", "a", " ", "z", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["\[Alpha]", "+", "1"]], ")"]]]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "\[Alpha]"]], "2"], ",", RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], " ", SuperscriptBox["z", "2"]]]]], "]"]]]], SqrtBox["\[Pi]"]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["8", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "4"], ",", FractionBox["5", "4"], ",", FractionBox["3", "2"]]], "}"]], ",", FractionBox[RowBox[List[SuperscriptBox["a", "4"], " ", SuperscriptBox["z", "4"]]], "4"]]], "]"]]]], "\[Pi]"], "+", RowBox[List[SqrtBox["2"], " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"], " ", RowBox[List["Gamma", "[", FractionBox[RowBox[List["2", "+", "\[Alpha]"]], "4"], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", FractionBox[RowBox[List["2", "+", "\[Alpha]"]], "4"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "4"], ",", FractionBox["5", "4"], ",", FractionBox[RowBox[List["6", "+", "\[Alpha]"]], "4"]]], "}"]], ",", FractionBox[RowBox[List[SuperscriptBox["a", "4"], " ", SuperscriptBox["z", "4"]]], "4"]]], "]"]]]]]], ")"]]]], RowBox[List["2", " ", "\[Alpha]"]]]]]]]

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

 2001-10-29