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FresnelC






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > FresnelC[z] > Integration > Indefinite integration > Involving direct function and Gamma-, Beta-, Erf-type functions > Involving erf-type functions > Involving erfi





http://functions.wolfram.com/06.33.21.0121.01









  


  










Input Form





Integrate[Erfi[b z] FresnelC[a z], z] == (1/(4 a b Pi)) ((1 - I) Sqrt[2 I b^2 + a^2 Pi] Erf[((1 + I)/2) Sqrt[2 I b^2 + a^2 Pi] z] + (1 - I) Sqrt[-2 I b^2 + a^2 Pi] Erfi[((1 + I)/2) Sqrt[-2 I b^2 + a^2 Pi] z] - 4 a E^(b^2 z^2) Sqrt[Pi] FresnelC[a z] + 4 a b Pi z Erfi[b z] FresnelC[a z] - 4 b Erfi[b z] Sin[(1/2) a^2 Pi z^2])










Standard Form





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MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[RowBox[List["Erfi", "[", RowBox[List["b_", " ", "z_"]], "]"]], " ", RowBox[List["FresnelC", "[", RowBox[List["a_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], " ", SqrtBox[RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", SuperscriptBox["b", "2"]]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]]]]], " ", RowBox[List["Erf", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "\[ImaginaryI]"]], ")"]], " ", SqrtBox[RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", SuperscriptBox["b", "2"]]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]]]]], " ", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], " ", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", SuperscriptBox["b", "2"]]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]]]]], " ", RowBox[List["Erfi", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "\[ImaginaryI]"]], ")"]], " ", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", SuperscriptBox["b", "2"]]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]]]]], " ", "z"]], "]"]]]], "-", RowBox[List["4", " ", "a", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox["z", "2"]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["FresnelC", "[", RowBox[List["a", " ", "z"]], "]"]]]], "+", RowBox[List["4", " ", "a", " ", "b", " ", "\[Pi]", " ", "z", " ", RowBox[List["Erfi", "[", RowBox[List["b", " ", "z"]], "]"]], " ", RowBox[List["FresnelC", "[", RowBox[List["a", " ", "z"]], "]"]]]], "-", RowBox[List["4", " ", "b", " ", RowBox[List["Erfi", "[", RowBox[List["b", " ", "z"]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox["a", "2"], " ", "\[Pi]", " ", SuperscriptBox["z", "2"]]], "]"]]]]]], RowBox[List["4", " ", "a", " ", "b", " ", "\[Pi]"]]]]]]]










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





2001-10-29