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http://functions.wolfram.com/06.33.21.0010.01
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Integrate[z^(\[Alpha] - 1) FresnelC[a z^r], z] ==
((1/(4 \[Alpha])) z^\[Alpha] (4 Pi^((r + \[Alpha])/(2 r))
(a^4 z^(4 r))^((r + \[Alpha])/(2 r)) FresnelC[a z^r] +
2^((r + \[Alpha])/(2 r)) a z^r ((I a^2 z^(2 r))^((r + \[Alpha])/(2 r))
Gamma[(r + \[Alpha])/(2 r), (-(1/2)) I a^2 Pi z^(2 r)] +
((-I) a^2 z^(2 r))^((r + \[Alpha])/(2 r)) Gamma[(r + \[Alpha])/(2 r),
(1/2) I a^2 Pi z^(2 r)])))/(Pi^((r + \[Alpha])/(2 r))
(a^4 z^(4 r))^((r + \[Alpha])/(2 r)))
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Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", RowBox[List["\[Alpha]", "-", "1"]]], RowBox[List["FresnelC", "[", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["4", " ", "\[Alpha]"]]], SuperscriptBox["\[Pi]", RowBox[List["-", FractionBox[RowBox[List["r", "+", "\[Alpha]"]], RowBox[List["2", " ", "r"]]]]]], " ", SuperscriptBox["z", "\[Alpha]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "4"], " ", SuperscriptBox["z", RowBox[List["4", " ", "r"]]]]], ")"]], RowBox[List["-", FractionBox[RowBox[List["r", "+", "\[Alpha]"]], RowBox[List["2", " ", "r"]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", SuperscriptBox["\[Pi]", FractionBox[RowBox[List["r", "+", "\[Alpha]"]], RowBox[List["2", " ", "r"]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "4"], " ", SuperscriptBox["z", RowBox[List["4", " ", "r"]]]]], ")"]], FractionBox[RowBox[List["r", "+", "\[Alpha]"]], RowBox[List["2", " ", "r"]]]], " ", RowBox[List["FresnelC", "[", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], "]"]]]], "+", RowBox[List[SuperscriptBox["2", FractionBox[RowBox[List["r", "+", "\[Alpha]"]], RowBox[List["2", " ", "r"]]]], " ", "a", " ", SuperscriptBox["z", "r"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]], ")"]], FractionBox[RowBox[List["r", "+", "\[Alpha]"]], RowBox[List["2", " ", "r"]]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["r", "+", "\[Alpha]"]], RowBox[List["2", " ", "r"]]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "\[Pi]", " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]], ")"]], FractionBox[RowBox[List["r", "+", "\[Alpha]"]], RowBox[List["2", " ", "r"]]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["r", "+", "\[Alpha]"]], RowBox[List["2", " ", "r"]]], ",", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "\[Pi]", " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]]]], "]"]]]]]], ")"]]]]]], ")"]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mo> ∫ </mo> <mrow> <mrow> <msup> <mi> z </mi> <mrow> <mi> α </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <semantics> <mi> C </mi> <annotation encoding='Mathematica'> TagBox["C", FresnelC] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> r </mi> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> α </mi> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mi> π </mi> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> r </mi> <mo> + </mo> <mi> α </mi> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> r </mi> </mrow> </mfrac> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> α </mi> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> a </mi> <mn> 4 </mn> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> r </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> r </mi> <mo> + </mo> <mi> α </mi> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> r </mi> </mrow> </mfrac> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mn> 2 </mn> <mfrac> <mrow> <mi> r </mi> <mo> + </mo> <mi> α </mi> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> r </mi> </mrow> </mfrac> </msup> <mo> ⁢ </mo> <mi> a </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> r </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mfrac> <mrow> <mi> r </mi> <mo> + </mo> <mi> α </mi> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> r </mi> </mrow> </mfrac> </msup> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> r </mi> <mo> + </mo> <mi> α </mi> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> r </mi> </mrow> </mfrac> <mo> , </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> r </mi> </mrow> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> r </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mfrac> <mrow> <mi> r </mi> <mo> + </mo> <mi> α </mi> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> r </mi> </mrow> </mfrac> </msup> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> r </mi> <mo> + </mo> <mi> α </mi> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> r </mi> </mrow> </mfrac> <mo> , </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> r </mi> </mrow> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> r </mi> </msup> </mrow> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <msup> <mi> π </mi> <mfrac> <mrow> <mi> r </mi> <mo> + </mo> <mi> α </mi> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> r </mi> </mrow> </mfrac> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> a </mi> <mn> 4 </mn> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> r </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mfrac> <mrow> <mi> r </mi> <mo> + </mo> <mi> α </mi> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> r </mi> </mrow> </mfrac> </msup> <mo> ⁢ </mo> <mrow> <semantics> <mi> C </mi> <annotation encoding='Mathematica'> TagBox["C", FresnelC] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> r </mi> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <power /> <ci> z </ci> <apply> <plus /> <ci> α </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> FresnelC </ci> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> z </ci> <ci> r </ci> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> α </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <pi /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> r </ci> <ci> α </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> r </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <ci> α </ci> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> 4 </cn> <ci> r </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> r </ci> <ci> α </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> r </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <times /> <apply> <plus /> <ci> r </ci> <ci> α </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> r </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> a </ci> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> r </ci> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> r </ci> <ci> α </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> r </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <times /> <apply> <plus /> <ci> r </ci> <ci> α </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> r </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <imaginaryi /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <pi /> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> r </ci> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> r </ci> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> r </ci> <ci> α </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> r </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <times /> <apply> <plus /> <ci> r </ci> <ci> α </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> r </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <imaginaryi /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <pi /> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> r </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <ci> r </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <pi /> <apply> <times /> <apply> <plus /> <ci> r </ci> <ci> α </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> r </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> 4 </cn> <ci> r </ci> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> r </ci> <ci> α </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> r </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> FresnelC </ci> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> z </ci> <ci> r </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
