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FresnelC






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > FresnelC[z] > Series representations > Residue representations





http://functions.wolfram.com/06.33.06.0010.01









  


  










Input Form





FresnelC[z] == ((Pi/Sqrt[2]) Sum[Residue[(1/(((1/2) E^(Pi (I/4)) Sqrt[Pi] z)^(4 s) (Gamma[s + 1] Gamma[1/4 - s] Gamma[1 - s]))) Gamma[s + 1/4], {s, -(1/4) - j}], {j, 0, Infinity}])/E^(I (Pi/4))










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["FresnelC", "[", "z", "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["\[Pi]", " "]], SqrtBox["2"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["\[Pi]", "/", "4"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], RowBox[List["Residue", "[", RowBox[List[RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[Pi]", " ", RowBox[List["\[ImaginaryI]", "/", "4"]]]]], " ", SqrtBox["\[Pi]"], " ", "z"]], ")"]], RowBox[List[RowBox[List["-", "4"]], "s"]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["s", "+", "1"]], "]"]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "4"], "-", "s"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "s"]], "]"]]]]], RowBox[List["Gamma", "[", RowBox[List["s", "+", FractionBox["1", "4"]]], "]"]]]], ",", RowBox[List["{", RowBox[List["s", ",", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], "-", "j"]]]], "}"]]]], "]"]]]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <semantics> <mi> C </mi> <annotation encoding='Mathematica'> TagBox[&quot;C&quot;, FresnelC] </annotation> </semantics> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> <mn> 4 </mn> </mfrac> </mrow> </msup> <mtext> </mtext> </mrow> <msqrt> <mn> 2 </mn> </msqrt> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mrow> <msub> <mi> res </mi> <mi> s </mi> </msub> <mo> ( </mo> <mrow> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mfrac> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> <mn> 4 </mn> </mfrac> </msup> <mo> &#8290; </mo> <msqrt> <mi> &#960; </mi> </msqrt> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mn> 4 </mn> </mrow> <mo> &#8290; </mo> <mi> s </mi> </mrow> </msup> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> s </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> - </mo> <mi> s </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> s </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> s </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> j </mi> </mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> FresnelC </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <pi /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <pi /> <imaginaryi /> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <apply> <ci> Subscript </ci> <ci> res </ci> <ci> s </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <pi /> <imaginaryi /> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -4 </cn> <ci> s </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> s </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> s </ci> <cn type='rational'> 1 <sep /> 4 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 4 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["FresnelC", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["\[Pi]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["-", "\[ImaginaryI]"]], ")"]], " ", "\[Pi]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], RowBox[List["Residue", "[", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "4"]], " ", SqrtBox["\[Pi]"], " ", "z"]], ")"]], RowBox[List[RowBox[List["-", "4"]], " ", "s"]]], " ", RowBox[List["Gamma", "[", RowBox[List["s", "+", FractionBox["1", "4"]]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["s", "+", "1"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "4"], "-", "s"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "s"]], "]"]]]]], ",", RowBox[List["{", RowBox[List["s", ",", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], "-", "j"]]]], "}"]]]], "]"]]]]]], SqrtBox["2"]]]]]]










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





2001-10-29