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 FresnelC

 http://functions.wolfram.com/06.33.21.0019.01

 Input Form

 Integrate[z^(\[Alpha] - 1) E^(b z^2) FresnelC[a z], z] == ((1/2) z^\[Alpha] FresnelC[a z] Gamma[\[Alpha]/2, 0, (-b) z^2])/ ((-b) z^2)^(\[Alpha]/2) + (I/((a^4 z^4)^(\[Alpha]/2) a)) Sum[((2^((1/2) (-3 + 2 k + \[Alpha])) Pi^((1/2) (-1 - 2 k - \[Alpha])) z^(-1 - 2 k + \[Alpha]) b^k)/(a^(4 k) ((2 k + \[Alpha]) k!))) (Sqrt[(-I) a^2 z^2] (I a^2 z^2)^(k + \[Alpha]/2) Gamma[k + (\[Alpha] + 1)/2, (-(1/2)) I a^2 Pi z^2] - ((-I) a^2 z^2)^(k + \[Alpha]/2) Sqrt[I a^2 z^2] Gamma[k + (\[Alpha] + 1)/2, (1/2) I a^2 Pi 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["FresnelC", "[", RowBox[List["a", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox["z", "\[Alpha]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List[RowBox[List["-", "\[Alpha]"]], "/", "2"]]], " ", RowBox[List["FresnelC", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["\[Alpha]", "2"], ",", "0", ",", RowBox[List[RowBox[List["-", "b"]], " ", SuperscriptBox["z", "2"]]]]], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "4"], " ", SuperscriptBox["z", "4"]]], ")"]], RowBox[List["-", FractionBox["\[Alpha]", "2"]]]]]], "a"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[" ", RowBox[List[SuperscriptBox["2", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "+", RowBox[List["2", " ", "k"]], "+", "\[Alpha]"]], ")"]]]]], " ", SuperscriptBox["\[Pi]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", " ", "k"]], "-", "\[Alpha]"]], ")"]]]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", " ", "k"]], "+", "\[Alpha]"]]], " ", SuperscriptBox["a", RowBox[List[RowBox[List["-", "4"]], "k"]]], SuperscriptBox["b", "k"], " "]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "k"]], "+", "\[Alpha]"]], ")"]], RowBox[List["k", "!"]]]]], RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["k", "+", FractionBox["\[Alpha]", "2"]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["k", "+", FractionBox[RowBox[List["\[Alpha]", "+", "1"]], "2"]]], ",", 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["k", "+", FractionBox["\[Alpha]", "2"]]]], " ", SqrtBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["k", "+", FractionBox[RowBox[List["\[Alpha]", "+", "1"]], "2"]]], ",", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "\[Pi]", " ", SuperscriptBox["z", "2"]]]]], "]"]]]]]], ")"]]]]]]]]]]]]]]

 MathML Form

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

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

 2001-10-29