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 Erf

 http://functions.wolfram.com/06.25.21.0104.01

 Input Form

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

 MathML Form

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

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

 2001-10-29