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 Erfi

 http://functions.wolfram.com/06.28.21.0137.01

 Input Form

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