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 FresnelC

 http://functions.wolfram.com/06.33.21.0024.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["Sin", "[", RowBox[List["b", " ", SuperscriptBox["z", "2"]]], "]"]], " ", RowBox[List["FresnelC", "[", RowBox[List["a", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["a", RowBox[List["4", "b"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], " ", "k"]]], " ", SuperscriptBox["\[Pi]", RowBox[List["2", " ", "k"]]], SuperscriptBox["a", RowBox[List["4", "k"]]], SuperscriptBox["b", RowBox[List[RowBox[List["-", "2"]], " ", "k"]]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["4", " ", "k"]], "+", "1"]], ")"]], " ", RowBox[List[RowBox[List["(", RowBox[List["2", " ", "k"]], ")"]], "!"]]]]], RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]], ",", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", SuperscriptBox["z", "2"]]]]], "]"]], "+", " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]], ",", RowBox[List["\[ImaginaryI]", " ", "b", " ", SuperscriptBox["z", "2"]]]]], "]"]]]], ")"]]]]]]]]]]]]

 MathML Form

 sin ( b z 2 ) C TagBox["C", FresnelC] ( a z ) z - a 4 b k = 0 2 - 2 k π 2 k a 4 k b - 2 k ( Γ ( 2 k + 1 , - b z 2 ) + Γ ( 2 k + 1 , b z 2 ) ) ( 4 k + 1 ) ( 2 k ) ! z b z 2 FresnelC a z -1 a 4 b -1 k 0 2 -2 k 2 k a 4 k b -2 k Gamma 2 k 1 -1 b z 2 Gamma 2 k 1 b z 2 4 k 1 2 k -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[RowBox[List["Sin", "[", RowBox[List["b_", " ", SuperscriptBox["z_", "2"]]], "]"]], " ", RowBox[List["FresnelC", "[", RowBox[List["a_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List["a", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], " ", "k"]]], " ", SuperscriptBox["\[Pi]", RowBox[List["2", " ", "k"]]], " ", SuperscriptBox["a", RowBox[List["4", " ", "k"]]], " ", SuperscriptBox["b", RowBox[List[RowBox[List["-", "2"]], " ", "k"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]], ",", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", SuperscriptBox["z", "2"]]]]], "]"]], "+", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]], ",", RowBox[List["\[ImaginaryI]", " ", "b", " ", SuperscriptBox["z", "2"]]]]], "]"]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["4", " ", "k"]], "+", "1"]], ")"]], " ", RowBox[List[RowBox[List["(", RowBox[List["2", " ", "k"]], ")"]], "!"]]]]]]]]], RowBox[List["4", " ", "b"]]]]]]]]]

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

 2001-10-29