html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 FresnelC

 http://functions.wolfram.com/06.33.21.0034.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", RowBox[List["\[Alpha]", "-", "1"]]], RowBox[List["Sin", "[", RowBox[List["b", " ", SuperscriptBox["z", "2"]]], "]"]], RowBox[List["FresnelC", "[", RowBox[List["a", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox["z", "\[Alpha]"], " ", RowBox[List["FresnelC", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List[RowBox[List["-", "\[Alpha]"]], "/", "2"]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["\[Alpha]", "2"], ",", "0", ",", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", SuperscriptBox["z", "2"]]]]], "]"]]]], "-", 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"2"], " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "\[Pi]", " ", SuperscriptBox["z", "2"]]]]], "]"]]]]]], ")"]]]], ")"]]]]]]]], ")"]]]], ")"]]]]]], ")"]]]]]]]]

 MathML Form

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

 Rule Form

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

 2001-10-29