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/06.33.21.0064.01

 Input Form

 Integrate[z E^(b z^2) Cos[c z^2] FresnelC[a z], z] == (1/(2 (b^2 + c^2))) ((-(1/8)) a Sqrt[Pi] z ((-I) c ((1 - (1 + I) FresnelC[((1 - I) Sqrt[I (2 I b + 2 c + a^2 Pi) z^ 2])/Sqrt[2 Pi]] - (1 - I) FresnelS[ ((1 - I) Sqrt[I (2 I b + 2 c + a^2 Pi) z^2])/Sqrt[2 Pi]])/ Sqrt[(-(b - I (c + (a^2 Pi)/2))) z^2] + (-1 + (1 + I) FresnelC[((1 - I) Sqrt[(-(b + I (c + (a^2 Pi)/2))) z^2])/ Sqrt[Pi]] + (1 - I) FresnelS[ ((1 - I) Sqrt[(-(b + I (c + (a^2 Pi)/2))) z^2])/Sqrt[Pi]])/ Sqrt[(-(b + I (c + (a^2 Pi)/2))) z^2]) + b ((-1 + (1 + I) FresnelC[((1 - I) Sqrt[I (2 I b + 2 c + a^2 Pi) z^2])/ Sqrt[2 Pi]] + (1 - I) FresnelS[ ((1 - I) Sqrt[I (2 I b + 2 c + a^2 Pi) z^2])/Sqrt[2 Pi]])/ Sqrt[(-(b - I (c + (a^2 Pi)/2))) z^2] + (-1 + (1 + I) FresnelC[((1 - I) Sqrt[(-(b + I (c + (a^2 Pi)/2))) z^2])/ Sqrt[Pi]] + (1 - I) FresnelS[ ((1 - I) Sqrt[(-(b + I (c + (a^2 Pi)/2))) z^2])/Sqrt[Pi]])/ Sqrt[(-(b + I (c + (a^2 Pi)/2))) z^2]) - I c ((-1 + (1 + I) FresnelC[((1 - I) Sqrt[(-(b + I (c - (a^2 Pi)/2))) z^ 2])/Sqrt[Pi]] + (1 - I) FresnelS[ ((1 - I) Sqrt[(-(b + I (c - (a^2 Pi)/2))) z^2])/Sqrt[Pi]])/ Sqrt[(-(b + I (c - (a^2 Pi)/2))) z^2] + (1 - (1 + I) FresnelC[((1 - I) Sqrt[(-(b + (1/2) I (-2 c + a^2 Pi))) z^ 2])/Sqrt[Pi]] - (1 - I) FresnelS[ ((1 - I) Sqrt[(-(b + (1/2) I (-2 c + a^2 Pi))) z^2])/Sqrt[Pi]])/ Sqrt[(-(b + (1/2) I (-2 c + a^2 Pi))) z^2]) + b ((-1 + (1 + I) FresnelC[((1 - I) Sqrt[(-(b + I (c - (a^2 Pi)/2))) z^ 2])/Sqrt[Pi]] + (1 - I) FresnelS[ ((1 - I) Sqrt[(-(b + I (c - (a^2 Pi)/2))) z^2])/Sqrt[Pi]])/ Sqrt[(-(b + I (c - (a^2 Pi)/2))) z^2] + (-1 + (1 + I) FresnelC[((1 - I) Sqrt[(-(b + (1/2) I (-2 c + a^2 Pi))) z^2])/Sqrt[Pi]] + (1 - I) FresnelS[ ((1 - I) Sqrt[(-(b + (1/2) I (-2 c + a^2 Pi))) z^2])/Sqrt[Pi]])/ Sqrt[(-(b + (1/2) I (-2 c + a^2 Pi))) z^2])) + E^(b z^2) FresnelC[a z] (b Cos[c z^2] + c Sin[c z^2]))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List["z", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["b", " ", SuperscriptBox["z", "2"]]]], RowBox[List["Cos", "[", RowBox[List["c", " ", SuperscriptBox["z", "2"]]], "]"]], " ", RowBox[List["FresnelC", "[", RowBox[List["a", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["2", " ", RowBox[List["(", RowBox[List[SuperscriptBox["b", "2"], "+", SuperscriptBox["c", "2"]]], ")"]]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "8"]]], " ", "a", " ", SqrtBox["\[Pi]"], " ", "z", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "-", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "\[ImaginaryI]"]], ")"]], " ", RowBox[List["FresnelC", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], " ", SqrtBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", "b"]], "+", RowBox[List["2", " ", "c"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], ")"]], " ", SuperscriptBox["z", "2"]]]]]], SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], " ", RowBox[List["FresnelS", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], " ", SqrtBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", "b"]], "+", RowBox[List["2", " ", "c"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], ")"]], " ", SuperscriptBox["z", "2"]]]]]], SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]], "]"]]]]]], ")"]], "/", RowBox[List["(", SqrtBox[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["c", "+", FractionBox[RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]], "2"]]], ")"]]]]]], ")"]]]], " ", SuperscriptBox["z", "2"]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "\[ImaginaryI]"]], ")"]], " ", RowBox[List["FresnelC", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], " ", SqrtBox[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["c", "+", FractionBox[RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]], "2"]]], ")"]]]]]], ")"]]]], " ", SuperscriptBox["z", "2"]]]]]], SqrtBox["\[Pi]"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], " ", RowBox[List["FresnelS", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], " ", SqrtBox[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["c", "+", FractionBox[RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]], "2"]]], ")"]]]]]], ")"]]]], " ", SuperscriptBox["z", "2"]]]]]], SqrtBox["\[Pi]"]], "]"]]]]]], ")"]], "/", RowBox[List["(", SqrtBox[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["c", "+", FractionBox[RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]], "2"]]], ")"]]]]]], ")"]]]], " ", SuperscriptBox["z", "2"]]]], ")"]]]]]], ")"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "\[ImaginaryI]"]], ")"]], " ", RowBox[List["FresnelC", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], " ", SqrtBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", "b"]], "+", RowBox[List["2", " ", "c"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], ")"]], " ", SuperscriptBox["z", "2"]]]]]], SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], " ", RowBox[List["FresnelS", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], " ", SqrtBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", "b"]], "+", RowBox[List["2", " ", "c"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], ")"]], " ", SuperscriptBox["z", "2"]]]]]], SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]], "]"]]]]]], ")"]], "/", RowBox[List["(", SqrtBox[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["c", "+", FractionBox[RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]], "2"]]], ")"]]]]]], ")"]]]], " ", SuperscriptBox["z", "2"]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "\[ImaginaryI]"]], ")"]], " ", RowBox[List["FresnelC", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], " ", SqrtBox[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["c", "+", FractionBox[RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]], "2"]]], ")"]]]]]], ")"]]]], " ", SuperscriptBox["z", "2"]]]]]], SqrtBox["\[Pi]"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], " ", RowBox[List["FresnelS", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], " ", SqrtBox[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["c", "+", FractionBox[RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]], "2"]]], ")"]]]]]], ")"]]]], " ", SuperscriptBox["z", "2"]]]]]], SqrtBox["\[Pi]"]], "]"]]]]]], ")"]], "/", RowBox[List["(", SqrtBox[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["c", "+", FractionBox[RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]], "2"]]], ")"]]]]]], ")"]]]], " ", SuperscriptBox["z", "2"]]]], ")"]]]]]], ")"]]]], "-", RowBox[List["\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "\[ImaginaryI]"]], ")"]], " ", RowBox[List["FresnelC", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], " ", SqrtBox[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["c", "-", FractionBox[RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]], "2"]]], ")"]]]]]], ")"]]]], " ", SuperscriptBox["z", "2"]]]]]], SqrtBox["\[Pi]"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], " ", RowBox[List["FresnelS", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], " ", SqrtBox[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["c", "-", FractionBox[RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]], "2"]]], ")"]]]]]], ")"]]]], " ", SuperscriptBox["z", "2"]]]]]], SqrtBox["\[Pi]"]], "]"]]]]]], ")"]], "/", RowBox[List["(", SqrtBox[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["c", "-", FractionBox[RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]], "2"]]], ")"]]]]]], ")"]]]], " ", SuperscriptBox["z", "2"]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "-", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "\[ImaginaryI]"]], ")"]], " ", RowBox[List["FresnelC", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], " ", SqrtBox[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["b", "+", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "c"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], ")"]]]]]], ")"]]]], " ", SuperscriptBox["z", "2"]]]]]], SqrtBox["\[Pi]"]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], " ", RowBox[List["FresnelS", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], " ", SqrtBox[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["b", "+", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "c"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], ")"]]]]]], ")"]]]], " ", SuperscriptBox["z", "2"]]]]]], SqrtBox["\[Pi]"]], "]"]]]]]], ")"]], "/", RowBox[List["(", SqrtBox[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["b", "+", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "c"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], ")"]]]]]], ")"]]]], " ", SuperscriptBox["z", "2"]]]], ")"]]]]]], ")"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "\[ImaginaryI]"]], ")"]], " ", RowBox[List["FresnelC", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], " ", SqrtBox[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["c", "-", FractionBox[RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]], "2"]]], ")"]]]]]], ")"]]]], " ", SuperscriptBox["z", "2"]]]]]], SqrtBox["\[Pi]"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], " ", RowBox[List["FresnelS", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], " ", SqrtBox[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["c", "-", FractionBox[RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]], "2"]]], ")"]]]]]], ")"]]]], " ", SuperscriptBox["z", "2"]]]]]], SqrtBox["\[Pi]"]], "]"]]]]]], ")"]], "/", RowBox[List["(", SqrtBox[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["c", "-", FractionBox[RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]], "2"]]], ")"]]]]]], ")"]]]], " ", SuperscriptBox["z", "2"]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "\[ImaginaryI]"]], ")"]], " ", RowBox[List["FresnelC", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], " ", SqrtBox[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["b", "+", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "c"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], ")"]]]]]], ")"]]]], " ", SuperscriptBox["z", "2"]]]]]], SqrtBox["\[Pi]"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], " ", RowBox[List["FresnelS", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], " ", SqrtBox[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["b", "+", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "c"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], ")"]]]]]], ")"]]]], " ", SuperscriptBox["z", "2"]]]]]], SqrtBox["\[Pi]"]], "]"]]]]]], ")"]], "/", RowBox[List["(", SqrtBox[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["b", "+", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "c"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], ")"]]]]]], ")"]]]], " ", SuperscriptBox["z", "2"]]]], ")"]]]]]], ")"]]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["b", " ", SuperscriptBox["z", "2"]]]], " ", RowBox[List["FresnelC", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["Cos", "[", RowBox[List["c", " ", SuperscriptBox["z", "2"]]], "]"]]]], "+", RowBox[List["c", " ", RowBox[List["Sin", "[", RowBox[List["c", " ", SuperscriptBox["z", "2"]]], "]"]]]]]], ")"]]]]]], ")"]]]]]]]]

 MathML Form

 z b z 2 cos ( c z 2 ) C TagBox["C", FresnelC] ( a z ) z 1 2 ( b 2 + c 2 ) ( b z 2 C TagBox["C", FresnelC] ( a z ) ( b cos ( c z 2 ) + c sin ( c z 2 ) ) - 1 8 a π z ( b ( ( ( 1 + ) C TagBox["C", FresnelC] ( ( 1 - ) - ( b + ( π a 2 2 + c ) ) z 2 π ) + ( 1 - ) S TagBox["S", FresnelS] ( ( 1 - ) - ( b + ( π a 2 2 + c ) ) z 2 π ) - 1 ) / ( - ( b + ( π a 2 2 + c ) ) z 2 ) + ( ( 1 + ) C TagBox["C", FresnelC] ( ( 1 - ) ( π a 2 + 2 c + 2 b ) z 2 2 π ) + ( 1 - ) S TagBox["S", FresnelS] ( ( 1 - ) ( π a 2 + 2 c + 2 b ) z 2 2 π ) - 1 ) / ( - ( b - ( π a 2 2 + c ) ) z 2 ) ) + c ( - ) ( ( ( 1 + ) C TagBox["C", FresnelC] ( ( 1 - ) - ( b + ( π a 2 2 + c ) ) z 2 π ) + ( 1 - ) S TagBox["S", FresnelS] ( ( 1 - ) - ( b + ( π a 2 2 + c ) ) z 2 π ) - 1 ) / ( - ( b + ( π a 2 2 + c ) ) z 2 ) + ( - ( 1 + ) C TagBox["C", FresnelC] ( ( 1 - ) ( π a 2 + 2 c + 2 b ) z 2 2 π ) - ( 1 - ) S TagBox["S", FresnelS] ( ( 1 - ) ( π a 2 + 2 c + 2 b ) z 2 2 π ) + 1 ) / ( - ( b - ( π a 2 2 + c ) ) z 2 ) ) + b ( ( ( 1 + ) C TagBox["C", FresnelC] ( ( 1 - ) - ( b + ( c - a 2 π 2 ) ) z 2 π ) + ( 1 - ) S TagBox["S", FresnelS] ( ( 1 - ) - ( b + ( c - a 2 π 2 ) ) z 2 π ) - 1 ) / ( - ( b + ( c - a 2 π 2 ) ) z 2 ) + ( ( 1 + ) C TagBox["C", FresnelC] ( ( 1 - ) - ( b + 1 2 ( a 2 π - 2 c ) ) z 2 π ) + ( 1 - ) S TagBox["S", FresnelS] ( ( 1 - ) - ( b + 1 2 ( a 2 π - 2 c ) ) z 2 π ) - 1 ) / ( - ( b + 1 2 ( a 2 π - 2 c ) ) z 2 ) ) - c ( ( ( 1 + ) C TagBox["C", FresnelC] ( ( 1 - ) - ( b + ( c - a 2 π 2 ) ) z 2 π ) + ( 1 - ) S TagBox["S", FresnelS] ( ( 1 - ) - ( b + ( c - a 2 π 2 ) ) z 2 π ) - 1 ) / ( - ( b + ( c - a 2 π 2 ) ) z 2 ) + ( - ( 1 + ) C TagBox["C", FresnelC] ( ( 1 - ) - ( b + 1 2 ( a 2 π - 2 c ) ) z 2 π ) - ( 1 - ) S TagBox["S", FresnelS] ( ( 1 - ) - ( b + 1 2 ( a 2 π - 2 c ) ) z 2 π ) + 1 ) / ( - ( b + 1 2 ( a 2 π - 2 c ) ) z 2 ) ) ) ) z z b z 2 c z 2 FresnelC a z 1 2 b 2 c 2 -1 b z 2 FresnelC a z b c z 2 c c z 2 -1 1 8 a 1 2 z b 1 FresnelC 1 -1 -1 b a 2 2 -1 c z 2 1 2 1 2 -1 1 -1 FresnelS 1 -1 -1 b a 2 2 -1 c z 2 1 2 1 2 -1 -1 -1 b a 2 2 -1 c z 2 1 2 -1 1 FresnelC 1 -1 a 2 2 c 2 b z 2 1 2 2 1 2 -1 1 -1 FresnelS 1 -1 a 2 2 c 2 b z 2 1 2 2 1 2 -1 -1 -1 b -1 a 2 2 -1 c z 2 1 2 -1 c -1 1 FresnelC 1 -1 -1 b a 2 2 -1 c z 2 1 2 1 2 -1 1 -1 FresnelS 1 -1 -1 b a 2 2 -1 c z 2 1 2 1 2 -1 -1 -1 b a 2 2 -1 c z 2 1 2 -1 -1 1 FresnelC 1 -1 a 2 2 c 2 b z 2 1 2 2 1 2 -1 -1 1 -1 FresnelS 1 -1 a 2 2 c 2 b z 2 1 2 2 1 2 -1 1 -1 b -1 a 2 2 -1 c z 2 1 2 -1 b 1 FresnelC 1 -1 -1 b c -1 a 2 2 -1 z 2 1 2 1 2 -1 1 -1 FresnelS 1 -1 -1 b c -1 a 2 2 -1 z 2 1 2 1 2 -1 -1 -1 b c -1 a 2 2 -1 z 2 1 2 -1 1 FresnelC 1 -1 -1 b 1 2 a 2 -1 2 c z 2 1 2 1 2 -1 1 -1 FresnelS 1 -1 -1 b 1 2 a 2 -1 2 c z 2 1 2 1 2 -1 -1 -1 b 1 2 a 2 -1 2 c z 2 1 2 -1 -1 c 1 FresnelC 1 -1 -1 b c -1 a 2 2 -1 z 2 1 2 1 2 -1 1 -1 FresnelS 1 -1 -1 b c -1 a 2 2 -1 z 2 1 2 1 2 -1 -1 -1 b c -1 a 2 2 -1 z 2 1 2 -1 -1 1 FresnelC 1 -1 -1 b 1 2 a 2 -1 2 c z 2 1 2 1 2 -1 -1 1 -1 FresnelS 1 -1 -1 b 1 2 a 2 -1 2 c z 2 1 2 1 2 -1 1 -1 b 1 2 a 2 -1 2 c z 2 1 2 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List["z_", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["b_", " ", SuperscriptBox["z_", "2"]]]], " ", RowBox[List["Cos", "[", RowBox[List["c_", " ", SuperscriptBox["z_", "2"]]], "]"]], " ", RowBox[List["FresnelC", "[", RowBox[List["a_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "8"]]], " ", "a", " ", SqrtBox["\[Pi]"], " ", "z", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "c", " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["1", "-", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "\[ImaginaryI]"]], ")"]], " ", RowBox[List["FresnelC", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], " ", SqrtBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", "b"]], "+", RowBox[List["2", " ", "c"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], ")"]], " ", SuperscriptBox["z", "2"]]]]]], SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], " ", RowBox[List["FresnelS", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], " ", SqrtBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", "b"]], "+", RowBox[List["2", " ", "c"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], ")"]], " ", SuperscriptBox["z", "2"]]]]]], SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]], "]"]]]]]], SqrtBox[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["c", "+", FractionBox[RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]], "2"]]], ")"]]]]]], ")"]]]], " ", SuperscriptBox["z", "2"]]]]], "+", FractionBox[RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "\[ImaginaryI]"]], ")"]], " ", RowBox[List["FresnelC", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], " ", SqrtBox[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["c", "+", FractionBox[RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]], "2"]]], ")"]]]]]], ")"]]]], " ", SuperscriptBox["z", "2"]]]]]], SqrtBox["\[Pi]"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], " ", RowBox[List["FresnelS", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], " ", SqrtBox[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["c", "+", FractionBox[RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]], "2"]]], ")"]]]]]], ")"]]]], " ", SuperscriptBox["z", "2"]]]]]], SqrtBox["\[Pi]"]], "]"]]]]]], SqrtBox[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["c", "+", FractionBox[RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]], "2"]]], ")"]]]]]], ")"]]]], " ", SuperscriptBox["z", "2"]]]]]]], ")"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "\[ImaginaryI]"]], ")"]], " ", RowBox[List["FresnelC", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], " ", SqrtBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", "b"]], "+", RowBox[List["2", " ", "c"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], ")"]], " ", SuperscriptBox["z", "2"]]]]]], SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], " ", RowBox[List["FresnelS", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], " ", SqrtBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", "b"]], "+", RowBox[List["2", " ", "c"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], ")"]], " ", SuperscriptBox["z", "2"]]]]]], SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]], "]"]]]]]], SqrtBox[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["c", "+", FractionBox[RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]], "2"]]], ")"]]]]]], ")"]]]], " ", SuperscriptBox["z", "2"]]]]], "+", FractionBox[RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "\[ImaginaryI]"]], ")"]], " ", RowBox[List["FresnelC", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], " ", SqrtBox[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["c", "+", FractionBox[RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]], "2"]]], ")"]]]]]], ")"]]]], " ", SuperscriptBox["z", "2"]]]]]], SqrtBox["\[Pi]"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], " ", RowBox[List["FresnelS", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], " ", SqrtBox[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["c", "+", FractionBox[RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]], "2"]]], ")"]]]]]], ")"]]]], " ", SuperscriptBox["z", "2"]]]]]], SqrtBox["\[Pi]"]], "]"]]]]]], SqrtBox[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["c", "+", FractionBox[RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]], "2"]]], ")"]]]]]], ")"]]]], " ", SuperscriptBox["z", "2"]]]]]]], ")"]]]], "-", RowBox[List["\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "\[ImaginaryI]"]], ")"]], " ", RowBox[List["FresnelC", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], " ", SqrtBox[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["c", "-", FractionBox[RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]], "2"]]], ")"]]]]]], ")"]]]], " ", SuperscriptBox["z", "2"]]]]]], SqrtBox["\[Pi]"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], " ", RowBox[List["FresnelS", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], " ", SqrtBox[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["c", "-", FractionBox[RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]], "2"]]], ")"]]]]]], ")"]]]], " ", SuperscriptBox["z", "2"]]]]]], SqrtBox["\[Pi]"]], "]"]]]]]], SqrtBox[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["c", "-", FractionBox[RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]], "2"]]], ")"]]]]]], ")"]]]], " ", SuperscriptBox["z", "2"]]]]], "+", FractionBox[RowBox[List["1", "-", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "\[ImaginaryI]"]], ")"]], " ", RowBox[List["FresnelC", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], " ", SqrtBox[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["b", "+", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "c"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], ")"]]]]]], ")"]]]], " ", SuperscriptBox["z", "2"]]]]]], SqrtBox["\[Pi]"]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], " ", RowBox[List["FresnelS", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], " ", SqrtBox[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["b", "+", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "c"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], ")"]]]]]], ")"]]]], " ", SuperscriptBox["z", "2"]]]]]], SqrtBox["\[Pi]"]], "]"]]]]]], SqrtBox[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["b", "+", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "c"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], ")"]]]]]], ")"]]]], " ", SuperscriptBox["z", "2"]]]]]]], ")"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "\[ImaginaryI]"]], ")"]], " ", RowBox[List["FresnelC", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], " ", SqrtBox[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["c", "-", FractionBox[RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]], "2"]]], ")"]]]]]], ")"]]]], " ", SuperscriptBox["z", "2"]]]]]], SqrtBox["\[Pi]"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], " ", RowBox[List["FresnelS", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], " ", SqrtBox[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["c", "-", FractionBox[RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]], "2"]]], ")"]]]]]], ")"]]]], " ", SuperscriptBox["z", "2"]]]]]], SqrtBox["\[Pi]"]], "]"]]]]]], SqrtBox[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["c", "-", FractionBox[RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]], "2"]]], ")"]]]]]], ")"]]]], " ", SuperscriptBox["z", "2"]]]]], "+", FractionBox[RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "\[ImaginaryI]"]], ")"]], " ", RowBox[List["FresnelC", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], " ", SqrtBox[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["b", "+", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "c"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], ")"]]]]]], ")"]]]], " ", SuperscriptBox["z", "2"]]]]]], SqrtBox["\[Pi]"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], " ", RowBox[List["FresnelS", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], " ", SqrtBox[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["b", "+", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "c"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], ")"]]]]]], ")"]]]], " ", SuperscriptBox["z", "2"]]]]]], SqrtBox["\[Pi]"]], "]"]]]]]], SqrtBox[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["b", "+", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "c"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], ")"]]]]]], ")"]]]], " ", SuperscriptBox["z", "2"]]]]]]], ")"]]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["b", " ", SuperscriptBox["z", "2"]]]], " ", RowBox[List["FresnelC", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["Cos", "[", RowBox[List["c", " ", SuperscriptBox["z", "2"]]], "]"]]]], "+", RowBox[List["c", " ", RowBox[List["Sin", "[", RowBox[List["c", " ", SuperscriptBox["z", "2"]]], "]"]]]]]], ")"]]]]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List[SuperscriptBox["b", "2"], "+", SuperscriptBox["c", "2"]]], ")"]]]]]]]]]

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

 2001-10-29