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 Erfc

 http://functions.wolfram.com/06.27.21.0106.01

 Input Form

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

 Standard Form

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

 MathML Form

 z α - 1 b z cosh ( c z ) erfc ( a z ) z 1 2 z α ( - ( ( c - b ) z ) - α Γ ( α , ( c - b ) z ) - ( - ( b + c ) z ) - α Γ ( α , - ( b + c ) z ) ) - a z α ( - ( b - c ) z ) - α ( b - c ) π k = 0 ( - 1 ) k a 2 k ( b - c ) - 2 k ( 2 k + 1 ) k ! Γ ( 2 k + α + 1 , - ( b - c ) z ) - a z α ( - ( b + c ) z ) - α ( b + c ) π k = 0 ( - 1 ) k a 2 k ( b + c ) - 2 k ( 2 k + 1 ) k ! Γ ( 2 k + α + 1 , - ( b + c ) z ) z z α -1 b z c z Erfc a z 1 2 z α -1 c -1 b z -1 α Gamma α c -1 b z -1 -1 b c z -1 α Gamma α -1 b c z -1 a z α -1 b -1 c z -1 α b -1 c 1 2 -1 k 0 -1 k a 2 k b -1 c -2 k 2 k 1 k -1 Gamma 2 k α 1 -1 b -1 c z -1 a z α -1 b c z -1 α b c 1 2 -1 k 0 -1 k a 2 k b c -2 k 2 k 1 k -1 Gamma 2 k α 1 -1 b c z [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", RowBox[List["\[Alpha]_", "-", "1"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["b_", " ", "z_"]]], " ", RowBox[List["Cosh", "[", RowBox[List["c_", " ", "z_"]], "]"]], " ", RowBox[List["Erfc", "[", RowBox[List["a_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox["z", "\[Alpha]"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", "c"]], ")"]], " ", "z"]], ")"]], RowBox[List["-", "\[Alpha]"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", ",", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", "c"]], ")"]], " ", "z"]]]], "]"]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]]]], " ", "z"]], ")"]], RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", ",", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]]]], " ", "z"]]]], "]"]]]]]], ")"]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "\[Alpha]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]]]], " ", "z"]], ")"]], RowBox[List["-", "\[Alpha]"]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["a", RowBox[List["2", " ", "k"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], RowBox[List[RowBox[List["-", "2"]], " ", "k"]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", RowBox[List["2", " ", "k"]], "+", "\[Alpha]"]], ",", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]]]], " ", "z"]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]], ")"]], " ", RowBox[List["k", "!"]]]]]]]]], RowBox[List[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], " ", SqrtBox["\[Pi]"]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "\[Alpha]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]]]], " ", "z"]], ")"]], RowBox[List["-", "\[Alpha]"]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["a", RowBox[List["2", " ", "k"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], RowBox[List[RowBox[List["-", "2"]], " ", "k"]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", RowBox[List["2", " ", "k"]], "+", "\[Alpha]"]], ",", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]]]], " ", "z"]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]], ")"]], " ", RowBox[List["k", "!"]]]]]]]]], RowBox[List[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], " ", SqrtBox["\[Pi]"]]]]]]]]]]

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

 2001-10-29