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 FresnelC

 http://functions.wolfram.com/06.33.21.0010.01

 Input Form

 Integrate[z^(\[Alpha] - 1) FresnelC[a z^r], z] == ((1/(4 \[Alpha])) z^\[Alpha] (4 Pi^((r + \[Alpha])/(2 r)) (a^4 z^(4 r))^((r + \[Alpha])/(2 r)) FresnelC[a z^r] + 2^((r + \[Alpha])/(2 r)) a z^r ((I a^2 z^(2 r))^((r + \[Alpha])/(2 r)) Gamma[(r + \[Alpha])/(2 r), (-(1/2)) I a^2 Pi z^(2 r)] + ((-I) a^2 z^(2 r))^((r + \[Alpha])/(2 r)) Gamma[(r + \[Alpha])/(2 r), (1/2) I a^2 Pi z^(2 r)])))/(Pi^((r + \[Alpha])/(2 r)) (a^4 z^(4 r))^((r + \[Alpha])/(2 r)))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", RowBox[List["\[Alpha]", "-", "1"]]], RowBox[List["FresnelC", "[", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["4", " ", "\[Alpha]"]]], SuperscriptBox["\[Pi]", RowBox[List["-", FractionBox[RowBox[List["r", "+", "\[Alpha]"]], RowBox[List["2", " ", "r"]]]]]], " ", SuperscriptBox["z", "\[Alpha]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "4"], " ", SuperscriptBox["z", RowBox[List["4", " ", "r"]]]]], ")"]], RowBox[List["-", FractionBox[RowBox[List["r", "+", "\[Alpha]"]], RowBox[List["2", " ", "r"]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", SuperscriptBox["\[Pi]", FractionBox[RowBox[List["r", "+", "\[Alpha]"]], RowBox[List["2", " ", "r"]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "4"], " ", SuperscriptBox["z", RowBox[List["4", " ", "r"]]]]], ")"]], FractionBox[RowBox[List["r", "+", "\[Alpha]"]], RowBox[List["2", " ", "r"]]]], " ", RowBox[List["FresnelC", "[", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], "]"]]]], "+", RowBox[List[SuperscriptBox["2", FractionBox[RowBox[List["r", "+", "\[Alpha]"]], RowBox[List["2", " ", "r"]]]], " ", "a", " ", SuperscriptBox["z", "r"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]], ")"]], FractionBox[RowBox[List["r", "+", "\[Alpha]"]], RowBox[List["2", " ", "r"]]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["r", "+", "\[Alpha]"]], RowBox[List["2", " ", "r"]]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "\[Pi]", " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]], ")"]], FractionBox[RowBox[List["r", "+", "\[Alpha]"]], RowBox[List["2", " ", "r"]]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["r", "+", "\[Alpha]"]], RowBox[List["2", " ", "r"]]], ",", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "\[Pi]", " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]]]], "]"]]]]]], ")"]]]]]], ")"]]]]]]]]

 MathML Form

 z α - 1 C TagBox["C", FresnelC] ( a z r ) z 1 4 α π - r + α 2 r z α ( a 4 z 4 r ) - r + α 2 r ( 2 r + α 2 r a ( ( - a 2 z 2 r ) r + α 2 r Γ ( r + α 2 r , 1 2 a 2 π z 2 r ) + ( a 2 z 2 r ) r + α 2 r Γ ( r + α 2 r , - 1 2 a 2 π z 2 r ) ) z r + 4 π r + α 2 r ( a 4 z 4 r ) r + α 2 r C TagBox["C", FresnelC] ( a z r ) ) z z α -1 FresnelC a z r 1 4 α -1 -1 r α 2 r -1 z α a 4 z 4 r -1 r α 2 r -1 2 r α 2 r -1 a -1 a 2 z 2 r r α 2 r -1 Gamma r α 2 r -1 1 2 a 2 z 2 r a 2 z 2 r r α 2 r -1 Gamma r α 2 r -1 -1 1 2 a 2 z 2 r z r 4 r α 2 r -1 a 4 z 4 r r α 2 r -1 FresnelC a z r [/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["FresnelC", "[", RowBox[List["a_", " ", SuperscriptBox["z_", "r_"]]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox["\[Pi]", RowBox[List["-", FractionBox[RowBox[List["r", "+", "\[Alpha]"]], RowBox[List["2", " ", "r"]]]]]], " ", SuperscriptBox["z", "\[Alpha]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "4"], " ", SuperscriptBox["z", RowBox[List["4", " ", "r"]]]]], ")"]], RowBox[List["-", FractionBox[RowBox[List["r", "+", "\[Alpha]"]], RowBox[List["2", " ", "r"]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", SuperscriptBox["\[Pi]", FractionBox[RowBox[List["r", "+", "\[Alpha]"]], RowBox[List["2", " ", "r"]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "4"], " ", SuperscriptBox["z", RowBox[List["4", " ", "r"]]]]], ")"]], FractionBox[RowBox[List["r", "+", "\[Alpha]"]], RowBox[List["2", " ", "r"]]]], " ", RowBox[List["FresnelC", "[", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], "]"]]]], "+", RowBox[List[SuperscriptBox["2", FractionBox[RowBox[List["r", "+", "\[Alpha]"]], RowBox[List["2", " ", "r"]]]], " ", "a", " ", SuperscriptBox["z", "r"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]], ")"]], FractionBox[RowBox[List["r", "+", "\[Alpha]"]], RowBox[List["2", " ", "r"]]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["r", "+", "\[Alpha]"]], RowBox[List["2", " ", "r"]]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "\[Pi]", " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]], ")"]], FractionBox[RowBox[List["r", "+", "\[Alpha]"]], RowBox[List["2", " ", "r"]]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["r", "+", "\[Alpha]"]], RowBox[List["2", " ", "r"]]], ",", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "\[Pi]", " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]]]], "]"]]]]]], ")"]]]]]], ")"]]]], RowBox[List["4", " ", "\[Alpha]"]]]]]]]

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

 2001-10-29