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 Erfc

 http://functions.wolfram.com/06.27.21.0133.01

 Input Form

 Integrate[(t^(\[Alpha] - 1) Erfc[t])/E^(z t), {t, 0, Infinity}] == (1/Sqrt[Pi]) ((1/\[Alpha]) Gamma[(1 + \[Alpha])/2] HypergeometricPFQ[{(\[Alpha] + 1)/2, \[Alpha]/2}, {1/2, 1 + \[Alpha]/2}, z^2/4] - (z/(1 + \[Alpha])) Gamma[1 + \[Alpha]/2] HypergeometricPFQ[{(\[Alpha] + 1)/2, 1 + \[Alpha]/2}, {3/2, (3 + \[Alpha])/2}, z^2/4]) /; Re[z] > 0 && Re[\[Alpha]] > 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[SuperscriptBox["t", RowBox[List["\[Alpha]", "-", "1"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "z"]], " ", "t"]]], RowBox[List["Erfc", "[", "t", "]"]], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List[SqrtBox["\[Pi]"], " "]]], RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", "\[Alpha]"], RowBox[List["Gamma", "[", FractionBox[RowBox[List["1", "+", "\[Alpha]"]], "2"], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["\[Alpha]", "+", "1"]], "2"], ",", FractionBox["\[Alpha]", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", RowBox[List["1", "+", FractionBox["\[Alpha]", "2"]]]]], "}"]], ",", FractionBox[SuperscriptBox["z", "2"], "4"]]], "]"]]]], "-", RowBox[List[FractionBox["z", RowBox[List["1", "+", "\[Alpha]"]]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", FractionBox["\[Alpha]", "2"]]], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["\[Alpha]", "+", "1"]], "2"], ",", RowBox[List["1", "+", FractionBox["\[Alpha]", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], ",", FractionBox[RowBox[List["3", "+", "\[Alpha]"]], "2"]]], "}"]], ",", FractionBox[SuperscriptBox["z", "2"], "4"]]], "]"]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", "z", "]"]], ">", "0"]], "\[And]", RowBox[List[RowBox[List["Re", "[", "\[Alpha]", "]"]], ">", "0"]]]]]]]]

 MathML Form

 0 t α - 1 - z t erfc ( t ) t 1 π ( 1 α Γ ( α + 1 2 ) 2 F 2 ( α + 1 2 , α 2 ; 1 2 , α 2 + 1 ; z 2 4 ) 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[FractionBox[RowBox[List["\[Alpha]", "+", "1"]], "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox["\[Alpha]", "2"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox["1", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Alpha]", "2"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[FractionBox[SuperscriptBox["z", "2"], "4"], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] - z α + 1 Γ ( α 2 + 1 ) 2 F 2 ( α + 1 2 , α 2 + 1 ; 3 2 , α + 3 2 ; z 2 4 ) 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[FractionBox[RowBox[List["\[Alpha]", "+", "1"]], "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Alpha]", "2"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox["3", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["\[Alpha]", "+", "3"]], "2"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[FractionBox[SuperscriptBox["z", "2"], "4"], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] ) /; Re ( z ) > 0 Re ( α ) > 0 Condition t 0 t α -1 -1 z t Erfc t 1 1 2 -1 1 α -1 Gamma α 1 2 -1 HypergeometricPFQ α 1 2 -1 α 2 -1 1 2 α 2 -1 1 z 2 4 -1 -1 z α 1 -1 Γ α 2 -1 1 HypergeometricPFQ α 1 2 -1 α 2 -1 1 3 2 α 3 2 -1 z 2 4 -1 z 0 α 0 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[RowBox[List[SuperscriptBox["t_", RowBox[List["\[Alpha]_", "-", "1"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "z_"]], " ", "t_"]]], " ", RowBox[List["Erfc", "[", "t_", "]"]]]], RowBox[List["\[DifferentialD]", "t_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", FractionBox[RowBox[List["1", "+", "\[Alpha]"]], "2"], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["\[Alpha]", "+", "1"]], "2"], ",", FractionBox["\[Alpha]", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", RowBox[List["1", "+", FractionBox["\[Alpha]", "2"]]]]], "}"]], ",", FractionBox[SuperscriptBox["z", "2"], "4"]]], "]"]]]], "\[Alpha]"], "-", FractionBox[RowBox[List["z", " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", FractionBox["\[Alpha]", "2"]]], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["\[Alpha]", "+", "1"]], "2"], ",", RowBox[List["1", "+", FractionBox["\[Alpha]", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], ",", FractionBox[RowBox[List["3", "+", "\[Alpha]"]], "2"]]], "}"]], ",", FractionBox[SuperscriptBox["z", "2"], "4"]]], "]"]]]], RowBox[List["1", "+", "\[Alpha]"]]]]], SqrtBox["\[Pi]"]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", "z", "]"]], ">", "0"]], "&&", RowBox[List[RowBox[List["Re", "[", "\[Alpha]", "]"]], ">", "0"]]]]]]]]]]

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

 2001-10-29