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 FresnelC

 http://functions.wolfram.com/06.33.21.0058.01

 Input Form

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

 MathML Form

 z α - 1 b z cos ( c z ) C TagBox["C", FresnelC] ( a z ) z a z α 2 k = 0 ( - 1 ) k 2 - 2 k π 2 k a 4 k ( 4 k + 1 ) ( 2 k ) ! ( ( b + c ) - 4 k - 1 ( ( - b - c ) z ) - α Γ ( 4 k + α + 1 , - ( b + c ) z ) + ( b - c ) - 4 k - 1 ( ( c - b ) z ) - α Γ ( 4 k + α + 1 , - ( b - c ) z ) ) z z α -1 b z c z FresnelC a z a z α 2 -1 k 0 -1 k 2 -2 k 2 k a 4 k 4 k 1 2 k -1 b c -4 k -1 -1 b -1 c z -1 α Gamma 4 k α 1 -1 b c z b -1 c -4 k -1 c -1 b z -1 α Gamma 4 k α 1 -1 b -1 c z [/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_", " ", "z_"]]], " ", RowBox[List["Cos", "[", RowBox[List["c_", " ", "z_"]], "]"]], " ", RowBox[List["FresnelC", "[", RowBox[List["a_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "\[Alpha]"]]], ")"]], " ", 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["-", "2"]], " ", "k"]]], " ", SuperscriptBox["\[Pi]", RowBox[List["2", " ", "k"]]], " ", SuperscriptBox["a", RowBox[List["4", " ", "k"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], " ", "z"]], ")"]], RowBox[List["-", "\[Alpha]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "k"]], "-", "1"]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", RowBox[List["4", " ", "k"]], "+", "\[Alpha]"]], ",", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]]]], " ", "z"]]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], " ", "z"]], ")"]], RowBox[List["-", "\[Alpha]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "k"]], "-", "1"]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", RowBox[List["4", " ", "k"]], "+", "\[Alpha]"]], ",", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]]]], " ", "z"]]]], "]"]]]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["4", " ", "k"]], "+", "1"]], ")"]], " ", RowBox[List[RowBox[List["(", RowBox[List["2", " ", "k"]], ")"]], "!"]]]]]]]]]]]]]

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

 2001-10-29