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 Erfc

 http://functions.wolfram.com/06.27.21.0130.01

 Input Form

 Integrate[z^(\[Alpha] - 1) Erf[b z] Erfc[a z], z] == (1/\[Alpha]) z^\[Alpha] (Erfc[a z] Erf[b z] + (b z (b^2 z^2)^((1/2) (-1 - \[Alpha])) Gamma[(1 + \[Alpha])/2, b^2 z^2])/ Sqrt[Pi]) - (((2 b z^\[Alpha])/(Pi \[Alpha] a)) Sum[(b^(2 k) Gamma[1 + k + \[Alpha]/2, a^2 z^2])/ ((-a^2)^k ((2 k + 1) k!)), {k, 0, Infinity}])/(a^2 z^2)^(\[Alpha]/2) - (((2 a z^\[Alpha])/(Pi \[Alpha] b)) Sum[(a^(2 k) Gamma[1 + k + \[Alpha]/2, b^2 z^2])/ ((-b^2)^k ((2 k + 1) k!)), {k, 0, Infinity}])/(b^2 z^2)^(\[Alpha]/2)

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", RowBox[List["\[Alpha]", "-", "1"]]], " ", RowBox[List["Erf", "[", RowBox[List["b", " ", "z"]], "]"]], RowBox[List["Erfc", "[", RowBox[List["a", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["1", "\[Alpha]"], SuperscriptBox["z", "\[Alpha]"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Erfc", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["Erf", "[", RowBox[List["b", " ", "z"]], "]"]]]], "+", FractionBox[RowBox[List["b", " ", "z", " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "\[Alpha]"]], ")"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "\[Alpha]"]], "2"], ",", RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox["z", "2"]]]]], "]"]]]], SqrtBox["\[Pi]"]]]], ")"]]]], "-", RowBox[List[FractionBox[RowBox[List["2", "b", " ", SuperscriptBox["z", "\[Alpha]"]]], RowBox[List["\[Pi]", " ", "\[Alpha]", " ", "a"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["-", FractionBox["\[Alpha]", "2"]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["a", "2"]]], ")"]], RowBox[List["-", "k"]]], " ", SuperscriptBox["b", RowBox[List["2", " ", "k"]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", "k", "+", FractionBox["\[Alpha]", "2"]]], ",", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "k"]], "+", "1"]], ")"]], " ", RowBox[List["k", "!"]]]]]]]]], "-", RowBox[List[FractionBox[RowBox[List["2", " ", "a", " ", SuperscriptBox["z", "\[Alpha]"]]], RowBox[List["\[Pi]", " ", "\[Alpha]", " ", "b"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["-", FractionBox["\[Alpha]", "2"]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox["a", RowBox[List["2", " ", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["b", "2"]]], ")"]], RowBox[List["-", "k"]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", "k", "+", FractionBox["\[Alpha]", "2"]]], ",", RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox["z", "2"]]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "k"]], "+", "1"]], ")"]], " ", RowBox[List["k", "!"]]]]]]]]]]]]]]]

 MathML Form

 z α - 1 erf ( b z ) erfc ( a z ) z - 2 b z α ( a 2 z 2 ) - α 2 π α a k = 0 ( - a 2 ) - k b 2 k Γ ( k + α 2 + 1 , a 2 z 2 ) ( 2 k + 1 ) k ! + z α α ( b z ( b 2 z 2 ) - α + 1 2 π Γ ( α + 1 2 , b 2 z 2 ) + erf ( b z ) erfc ( a z ) ) - 2 a z α π α b ( b 2 z 2 ) - α 2 k = 0 a 2 k ( - b 2 ) - k Γ ( k + α 2 + 1 , b 2 z 2 ) ( 2 k + 1 ) k ! z z α -1 Erf b z Erfc a z -1 2 b z α a 2 z 2 -1 α 2 -1 α a -1 k 0 -1 a 2 -1 k b 2 k Gamma k α 2 -1 1 a 2 z 2 2 k 1 k -1 z α α -1 b z b 2 z 2 -1 α 1 2 -1 1 2 -1 Gamma α 1 2 -1 b 2 z 2 Erf b z Erfc a z -1 2 a z α α b -1 b 2 z 2 -1 α 2 -1 k 0 a 2 k -1 b 2 -1 k Gamma k α 2 -1 1 b 2 z 2 2 k 1 k -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["Erf", "[", RowBox[List["b_", " ", "z_"]], "]"]], " ", RowBox[List["Erfc", "[", RowBox[List["a_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["z", "\[Alpha]"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Erfc", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["Erf", "[", RowBox[List["b", " ", "z"]], "]"]]]], "+", FractionBox[RowBox[List["b", " ", "z", " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "\[Alpha]"]], ")"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "\[Alpha]"]], "2"], ",", RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox["z", "2"]]]]], "]"]]]], SqrtBox["\[Pi]"]]]], ")"]]]], "\[Alpha]"], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", "b", " ", SuperscriptBox["z", "\[Alpha]"]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["-", FractionBox["\[Alpha]", "2"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["a", "2"]]], ")"]], RowBox[List["-", "k"]]], " ", SuperscriptBox["b", RowBox[List["2", " ", "k"]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", "k", "+", FractionBox["\[Alpha]", "2"]]], ",", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]], " ", RowBox[List["k", "!"]]]]]]]]], RowBox[List["\[Pi]", " ", "\[Alpha]", " ", "a"]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", "a", " ", SuperscriptBox["z", "\[Alpha]"]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["-", FractionBox["\[Alpha]", "2"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox["a", RowBox[List["2", " ", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["b", "2"]]], ")"]], RowBox[List["-", "k"]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", "k", "+", FractionBox["\[Alpha]", "2"]]], ",", RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox["z", "2"]]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]], " ", RowBox[List["k", "!"]]]]]]]]], RowBox[List["\[Pi]", " ", "\[Alpha]", " ", "b"]]]]]]]]]

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

 2001-10-29