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 FresnelC

 http://functions.wolfram.com/06.33.21.0122.01

 Input Form

 Integrate[z^(\[Alpha] - 1) Erf[b z] FresnelC[a z], z] == (z^\[Alpha]/(Sqrt[Pi] \[Alpha])) FresnelC[a z] (Sqrt[Pi] Erf[b z] - b z (b^2 z^2)^((1/2) (-1 - \[Alpha])) Gamma[(1 + \[Alpha])/2, 0, b^2 z^2]) - 2 a Sum[(((-1)^k 2^(-1 + k + \[Alpha]/2) b^(1 + 2 k) Pi^(-k - (\[Alpha] + 3)/2) z^(2 + 2 k + \[Alpha]))/ ((1 + 2 k) (1 + 2 k + \[Alpha]) k!)) ((I a^2 z^2)^(-1 + k + \[Alpha]/2) (a^4 z^4)^(-k - \[Alpha]/2) Gamma[\[Alpha]/2 + k + 1, (-(1/2)) I a^2 Pi z^2] + ((-I) a^2 z^2)^(-1 + k + \[Alpha]/2) (a^4 z^4)^(-k - \[Alpha]/2) Gamma[\[Alpha]/2 + k + 1, (1/2) I a^2 Pi z^2]), {k, 0, Infinity}]

 Standard Form

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 MathML Form

 z α - 1 erf ( b z ) C TagBox["C", FresnelC] ( a z ) z z α C TagBox["C", FresnelC] ( a z ) π α ( π erf ( b z ) - b z ( b 2 z 2 ) 1 2 ( - α - 1 ) Γ ( α + 1 2 , 0 , b 2 z 2 ) ) - 2 a k = 0 ( - 1 ) k 2 k + α 2 - 1 b 2 k + 1 π - 1 2 ( α + 3 ) - k z 2 k + α + 2 ( 2 k + 1 ) ( 2 k + α + 1 ) k ! ( ( - a 2 z 2 ) k + α 2 - 1 ( a 4 z 4 ) - k - α 2 Γ ( k + α 2 + 1 , 1 2 a 2 π z 2 ) + ( a 2 z 2 ) k + α 2 - 1 ( a 4 z 4 ) - k - α 2 Γ ( k + α 2 + 1 , - 1 2 a 2 π z 2 ) ) z z α -1 Erf b z FresnelC a z z α FresnelC a z 1 2 α -1 1 2 Erf b z -1 b z b 2 z 2 1 2 -1 α -1 Gamma α 1 2 -1 0 b 2 z 2 -1 2 a k 0 -1 k 2 k α 2 -1 -1 b 2 k 1 -1 1 2 α 3 -1 k z 2 k α 2 2 k 1 2 k α 1 k -1 -1 a 2 z 2 k α 2 -1 -1 a 4 z 4 -1 k -1 α 2 -1 Gamma k α 2 -1 1 1 2 a 2 z 2 a 2 z 2 k α 2 -1 -1 a 4 z 4 -1 k -1 α 2 -1 Gamma k α 2 -1 1 -1 1 2 a 2 z 2 [/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["Erf", "[", RowBox[List["b_", " ", "z_"]], "]"]], " ", RowBox[List["FresnelC", "[", RowBox[List["a_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["z", "\[Alpha]"], " ", RowBox[List["FresnelC", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Erf", "[", RowBox[List["b", " ", "z"]], "]"]]]], "-", RowBox[List["b", " ", "z", " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["b", "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"], ",", "0", ",", RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox["z", "2"]]]]], "]"]]]]]], ")"]]]], RowBox[List[SqrtBox["\[Pi]"], " ", "\[Alpha]"]]], "-", RowBox[List["2", " ", "a", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "+", "k", "+", FractionBox["\[Alpha]", "2"]]]], " ", SuperscriptBox["b", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]]], " ", SuperscriptBox["\[Pi]", RowBox[List[RowBox[List["-", "k"]], "-", FractionBox[RowBox[List["\[Alpha]", "+", "3"]], "2"]]]], " ", SuperscriptBox["z", RowBox[List["2", "+", RowBox[List["2", " ", "k"]], "+", "\[Alpha]"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List[RowBox[List["-", "1"]], "+", "k", "+", FractionBox["\[Alpha]", "2"]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "4"], " ", SuperscriptBox["z", "4"]]], ")"]], RowBox[List[RowBox[List["-", "k"]], "-", FractionBox["\[Alpha]", "2"]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["\[Alpha]", "2"], "+", "k", "+", "1"]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "\[Pi]", " ", SuperscriptBox["z", "2"]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List[RowBox[List["-", "1"]], "+", "k", "+", FractionBox["\[Alpha]", "2"]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "4"], " ", SuperscriptBox["z", "4"]]], ")"]], RowBox[List[RowBox[List["-", "k"]], "-", FractionBox["\[Alpha]", "2"]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["\[Alpha]", "2"], "+", "k", "+", "1"]], ",", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "\[Pi]", " ", SuperscriptBox["z", "2"]]]]], "]"]]]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "k"]], "+", "\[Alpha]"]], ")"]], " ", RowBox[List["k", "!"]]]]]]]]]]]]]]]

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

 2001-10-29