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 Erfc

 http://functions.wolfram.com/06.27.21.0125.01

 Input Form

 Integrate[z^(\[Alpha] - 1) Erfc[a z] Erfc[b z], z] == (z^\[Alpha] Erfc[a z] Erfc[b z])/\[Alpha] - (a z^(1 + \[Alpha]) (a^2 z^2)^(-(1/2) - \[Alpha]/2) Gamma[(1 + \[Alpha])/2, a^2 z^2])/(Sqrt[Pi] \[Alpha]) - (b z^(1 + \[Alpha]) (b^2 z^2)^(-(1/2) - \[Alpha]/2) Gamma[(1 + \[Alpha])/2, b^2 z^2])/(Sqrt[Pi] \[Alpha]) + ((1/(a Pi \[Alpha])) 2 b z^\[Alpha] Sum[((-1)^k b^(2 k) Gamma[\[Alpha]/2 + k + 1, a^2 z^2])/ (a^(2 k) ((1 + 2 k) k!)), {k, 0, Infinity}])/(a^2 z^2)^(\[Alpha]/2) + ((1/(b Pi \[Alpha])) 2 a z^\[Alpha] Sum[((-1)^k a^(2 k) Gamma[\[Alpha]/2 + k + 1, b^2 z^2])/ (b^(2 k) ((1 + 2 k) 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["Erfc", "[", RowBox[List["a", " ", "z"]], "]"]], RowBox[List["Erfc", "[", RowBox[List["b", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["z", "\[Alpha]"], " ", RowBox[List["Erfc", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["Erfc", "[", RowBox[List["b", " ", "z"]], "]"]]]], "\[Alpha]"], "-", FractionBox[RowBox[List["a", " ", SuperscriptBox["z", RowBox[List["1", "+", "\[Alpha]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", FractionBox["\[Alpha]", "2"]]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "\[Alpha]"]], "2"], ",", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]]], "]"]]]], RowBox[List[SqrtBox["\[Pi]"], " ", "\[Alpha]"]]], "-", FractionBox[RowBox[List["b", " ", SuperscriptBox["z", RowBox[List["1", "+", "\[Alpha]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", FractionBox["\[Alpha]", "2"]]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "\[Alpha]"]], "2"], ",", RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox["z", "2"]]]]], "]"]]]], RowBox[List[SqrtBox["\[Pi]"], " ", "\[Alpha]"]]], "+", RowBox[List[FractionBox["1", RowBox[List["a", " ", "\[Pi]", " ", "\[Alpha]"]]], "2", " ", "b", " ", SuperscriptBox["z", "\[Alpha]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List[RowBox[List["-", "\[Alpha]"]], "/", "2"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["a", RowBox[List[RowBox[List["-", "2"]], " ", "k"]]], " ", SuperscriptBox["b", RowBox[List["2", " ", "k"]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["\[Alpha]", "2"], "+", "k", "+", "1"]], ",", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]], ")"]], " ", RowBox[List["k", "!"]]]]]]]]], "+", RowBox[List[FractionBox["1", RowBox[List["b", " ", "\[Pi]", " ", "\[Alpha]"]]], "2", " ", "a", " ", SuperscriptBox["z", "\[Alpha]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List[RowBox[List["-", "\[Alpha]"]], "/", "2"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["a", RowBox[List["2", " ", "k"]]], " ", SuperscriptBox["b", RowBox[List[RowBox[List["-", "2"]], " ", "k"]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["\[Alpha]", "2"], "+", "k", "+", "1"]], ",", RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox["z", "2"]]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]], ")"]], " ", RowBox[List["k", "!"]]]]]]]]]]]]]]]

 MathML Form

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

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

 2001-10-29