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 Erfc

 http://functions.wolfram.com/06.27.21.0115.01

 Input Form

 Integrate[z^(\[Alpha] - 1) Log[b z] Erfc[a z], z] == (z^\[Alpha] (a^2 z^2)^((1/2) (-1 - \[Alpha])) (-2 a z (a^2 z^2)^((1 + \[Alpha])/2) \[Alpha] HypergeometricPFQ[ {1/2 + \[Alpha]/2, 1/2 + \[Alpha]/2}, {3/2 + \[Alpha]/2, 3/2 + \[Alpha]/2}, (-a^2) z^2] + (1 + \[Alpha])^2 (Sqrt[Pi] (a^2 z^2)^((1 + \[Alpha])/2) Erfc[a z] (-1 + \[Alpha] Log[b z]) + a z (\[Alpha] Gamma[(1 + \[Alpha])/2] Log[z] + Gamma[(1 + \[Alpha])/2, a^2 z^2] (1 - \[Alpha] Log[b z])))))/(Sqrt[Pi] \[Alpha]^2 (1 + \[Alpha])^2)

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", RowBox[List["\[Alpha]", "-", "1"]]], " ", RowBox[List["Log", "[", RowBox[List["b", " ", "z"]], "]"]], RowBox[List["Erfc", "[", RowBox[List["a", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["z", "\[Alpha]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "\[Alpha]"]], ")"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "a", " ", "z", " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]], ")"]], FractionBox[RowBox[List["1", "+", "\[Alpha]"]], "2"]], " ", "\[Alpha]", " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", FractionBox["\[Alpha]", "2"]]], ",", RowBox[List[FractionBox["1", "2"], "+", FractionBox["\[Alpha]", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", FractionBox["\[Alpha]", "2"]]], ",", RowBox[List[FractionBox["3", "2"], "+", FractionBox["\[Alpha]", "2"]]]]], "}"]], ",", RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], " ", SuperscriptBox["z", "2"]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "\[Alpha]"]], ")"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox["\[Pi]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]], ")"]], FractionBox[RowBox[List["1", "+", "\[Alpha]"]], "2"]], " ", RowBox[List["Erfc", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["\[Alpha]", " ", RowBox[List["Log", "[", RowBox[List["b", " ", "z"]], "]"]]]]]], ")"]]]], "+", RowBox[List["a", " ", "z", " ", RowBox[List["(", RowBox[List[RowBox[List["\[Alpha]", " ", RowBox[List["Gamma", "[", FractionBox[RowBox[List["1", "+", "\[Alpha]"]], "2"], "]"]], " ", RowBox[List["Log", "[", "z", "]"]]]], "+", RowBox[List[RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "\[Alpha]"]], "2"], ",", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["\[Alpha]", " ", RowBox[List["Log", "[", RowBox[List["b", " ", "z"]], "]"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List[SqrtBox["\[Pi]"], " ", SuperscriptBox["\[Alpha]", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "\[Alpha]"]], ")"]], "2"]]], ")"]]]]]]]]

 MathML Form

 z α - 1 log ( b z ) erfc ( a z ) z z α ( a 2 z 2 ) 1 2 ( - α - 1 ) π α 2 ( α + 1 ) 2 ( ( α + 1 ) 2 ( π erfc ( a z ) ( α log ( b z ) - 1 ) ( a 2 z 2 ) α + 1 2 + a z ( α Γ ( α + 1 2 ) log ( z ) + Γ ( α + 1 2 , a 2 z 2 ) ( 1 - α log ( b z ) ) ) ) - 2 a z ( a 2 z 2 ) α + 1 2 α 2 F 2 ( α 2 + 1 2 , α 2 + 1 2 ; α 2 + 3 2 , α 2 + 3 2 ; - a 2 z 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox["\[Alpha]", "2"], "+", FractionBox["1", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Alpha]", "2"], "+", FractionBox["1", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox["\[Alpha]", "2"], "+", FractionBox["3", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Alpha]", "2"], "+", FractionBox["3", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], " ", SuperscriptBox["z", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] ) z z α -1 b z Erfc a z z α a 2 z 2 1 2 -1 α -1 1 2 α 2 α 1 2 -1 α 1 2 1 2 Erfc a z α b z -1 a 2 z 2 α 1 2 -1 a z α Gamma α 1 2 -1 z Gamma α 1 2 -1 a 2 z 2 1 -1 α b z -1 2 a z a 2 z 2 α 1 2 -1 α HypergeometricPFQ α 2 -1 1 2 α 2 -1 1 2 α 2 -1 3 2 α 2 -1 3 2 -1 a 2 z 2 [/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["Log", "[", RowBox[List["b_", " ", "z_"]], "]"]], " ", RowBox[List["Erfc", "[", RowBox[List["a_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox["z", "\[Alpha]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "\[Alpha]"]], ")"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "a", " ", "z", " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]], ")"]], FractionBox[RowBox[List["1", "+", "\[Alpha]"]], "2"]], " ", "\[Alpha]", " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", FractionBox["\[Alpha]", "2"]]], ",", RowBox[List[FractionBox["1", "2"], "+", FractionBox["\[Alpha]", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", FractionBox["\[Alpha]", "2"]]], ",", RowBox[List[FractionBox["3", "2"], "+", FractionBox["\[Alpha]", "2"]]]]], "}"]], ",", RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], " ", SuperscriptBox["z", "2"]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "\[Alpha]"]], ")"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox["\[Pi]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]], ")"]], FractionBox[RowBox[List["1", "+", "\[Alpha]"]], "2"]], " ", RowBox[List["Erfc", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["\[Alpha]", " ", RowBox[List["Log", "[", RowBox[List["b", " ", "z"]], "]"]]]]]], ")"]]]], "+", RowBox[List["a", " ", "z", " ", RowBox[List["(", RowBox[List[RowBox[List["\[Alpha]", " ", RowBox[List["Gamma", "[", FractionBox[RowBox[List["1", "+", "\[Alpha]"]], "2"], "]"]], " ", RowBox[List["Log", "[", "z", "]"]]]], "+", RowBox[List[RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "\[Alpha]"]], "2"], ",", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["\[Alpha]", " ", RowBox[List["Log", "[", RowBox[List["b", " ", "z"]], "]"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]], RowBox[List[SqrtBox["\[Pi]"], " ", SuperscriptBox["\[Alpha]", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "\[Alpha]"]], ")"]], "2"]]]]]]]]

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

 2001-10-29