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http://functions.wolfram.com/08.01.19.0002.01
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Im[EllipticE[x + I y]] ==
(-((Pi y)/8)) HypergeometricPFQ[{{3/4, 5/4, 1/4, 3/4}, {}, {}},
{{1, 3/2}, {3/2}, {1/2}}, -y^2, x^2] - ((3 Pi x y)/64)
HypergeometricPFQ[{{5/4, 7/4, 3/4, 5/4}, {}, {}},
{{2, 3/2}, {3/2}, {3/2}}, -y^2, x^2] /;
Element[x, Reals] && Element[y, Reals]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Im", "[", RowBox[List["EllipticE", "[", RowBox[List["x", "+", RowBox[List["\[ImaginaryI]", " ", "y"]]]], "]"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[Pi]", " ", "y"]], "8"]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["3", "4"], ",", FractionBox["5", "4"], ",", FractionBox["1", "4"], ",", FractionBox["3", "4"]]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["1", ",", FractionBox["3", "2"]]], "}"]], ",", RowBox[List["{", FractionBox["3", "2"], "}"]], ",", RowBox[List["{", FractionBox["1", "2"], "}"]]]], "}"]], ",", RowBox[List["-", SuperscriptBox["y", "2"]]], ",", SuperscriptBox["x", "2"]]], "]"]]]], "-", RowBox[List[FractionBox[RowBox[List["3", "\[Pi]", " ", "x", " ", "y"]], "64"], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["5", "4"], ",", FractionBox["7", "4"], ",", FractionBox["3", "4"], ",", FractionBox["5", "4"]]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["2", ",", FractionBox["3", "2"]]], "}"]], ",", RowBox[List["{", FractionBox["3", "2"], "}"]], ",", RowBox[List["{", FractionBox["3", "2"], "}"]]]], "}"]], ",", RowBox[List["-", SuperscriptBox["y", "2"]]], ",", SuperscriptBox["x", "2"]]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List["x", "\[Element]", "Reals"]], "\[And]", RowBox[List["y", "\[Element]", "Reals"]]]]]]]]
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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> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> E </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> x </mi> <mo> + </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> y </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> y </mi> </mrow> <mn> 8 </mn> </mfrac> </mrow> <mo> ⁢ </mo> <mrow> <msubsup> <mi> F </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mn> 1 </mn> <mo> ⁢ </mo> <mn> 1 </mn> </mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mn> 0 </mn> <mo> ⁢ </mo> <mn> 0 </mn> </mrow> </msubsup> <mo> ( </mo> <mrow> <mrow> <mtable> <mtr> <mtd> <mrow> <mfrac> <mn> 3 </mn> <mn> 4 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 5 </mn> <mn> 4 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> , </mo> <mrow> <mrow> <mrow> <mfrac> <mn> 3 </mn> <mn> 4 </mn> </mfrac> <mo> ; </mo> </mrow> <mo> ; </mo> </mrow> <mo> ; </mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn> 1 </mn> <mo> , </mo> <mrow> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> <mo> ; </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> <mo> ; </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ; </mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo> - </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> <mo> , </mo> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> x </mi> <mo> ⁢ </mo> <mi> y </mi> </mrow> <mn> 64 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <msubsup> <mi> F </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mn> 1 </mn> <mo> ⁢ </mo> <mn> 1 </mn> </mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mn> 0 </mn> <mo> ⁢ </mo> <mn> 0 </mn> </mrow> </msubsup> <mo> ( </mo> <mrow> <mrow> <mtable> <mtr> <mtd> <mrow> <mfrac> <mn> 5 </mn> <mn> 4 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 7 </mn> <mn> 4 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 3 </mn> <mn> 4 </mn> </mfrac> <mo> , </mo> <mrow> <mrow> <mrow> <mfrac> <mn> 5 </mn> <mn> 4 </mn> </mfrac> <mo> ; </mo> </mrow> <mo> ; </mo> </mrow> <mo> ; </mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn> 2 </mn> <mo> , </mo> <mrow> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> <mo> ; </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> <mo> ; </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> <mo> ; </mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo> - </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> <mo> , </mo> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> x </mi> <mo> ∈ </mo> <semantics> <mi> ℝ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalR]", Function[Reals]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> y </mi> <mo> ∈ </mo> <semantics> <mi> ℝ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalR]", Function[Reals]] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> FormBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> Im </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> E </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> x </ms> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <ms> ⅈ </ms> <ms> y </ms> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <ms> ) </ms> </list> </apply> <ms> ⩵ </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> π </ms> <ms> y </ms> </list> </apply> <ms> 8 </ms> </apply> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubsuperscriptBox </ci> <ms> F </ms> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> 1 </ms> <ms> 1 </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> 4 </ms> <ms> 0 </ms> <ms> 0 </ms> </list> </apply> </apply> <ms> [ </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> GridBox </ci> <list> <list> <apply> <ci> ErrorBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> 3 </ms> <ms> 4 </ms> </apply> <ms> , </ms> <apply> <ci> FractionBox </ci> <ms> 5 </ms> <ms> 4 </ms> </apply> <ms> , </ms> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 4 </ms> </apply> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> 3 </ms> <ms> 4 </ms> </apply> <ms> ; </ms> </list> </apply> <ms> ; </ms> </list> </apply> <ms> ; </ms> </list> </apply> </list> </apply> </apply> </list> <list> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> 3 </ms> <ms> 2 </ms> </apply> <ms> ; </ms> <apply> <ci> FractionBox </ci> <ms> 3 </ms> <ms> 2 </ms> </apply> <ms> ; </ms> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 2 </ms> </apply> <ms> ; </ms> </list> </apply> </list> </apply> </list> </list> </apply> <ms> - </ms> <apply> <ci> SuperscriptBox </ci> <ms> y </ms> <ms> 2 </ms> </apply> </list> </apply> <ms> , </ms> <apply> <ci> SuperscriptBox </ci> <ms> x </ms> <ms> 2 </ms> </apply> <ms> ] </ms> </list> </apply> </list> </apply> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> 3 </ms> <ms> π </ms> <ms> x </ms> <ms> y </ms> </list> </apply> <ms> 64 </ms> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubsuperscriptBox </ci> <ms> F </ms> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> 1 </ms> <ms> 1 </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> 4 </ms> <ms> 0 </ms> <ms> 0 </ms> </list> </apply> </apply> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> GridBox </ci> <list> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> 5 </ms> <ms> 4 </ms> </apply> <ms> , </ms> <apply> <ci> FractionBox </ci> <ms> 7 </ms> <ms> 4 </ms> </apply> <ms> , </ms> <apply> <ci> FractionBox </ci> <ms> 3 </ms> <ms> 4 </ms> </apply> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> 5 </ms> <ms> 4 </ms> </apply> <ms> ; </ms> </list> </apply> <ms> ; </ms> </list> </apply> <ms> ; </ms> </list> </apply> </list> </apply> </list> <list> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> 3 </ms> <ms> 2 </ms> </apply> <ms> ; </ms> <apply> <ci> FractionBox </ci> <ms> 3 </ms> <ms> 2 </ms> </apply> <ms> ; </ms> <apply> <ci> FractionBox </ci> <ms> 3 </ms> <ms> 2 </ms> </apply> <ms> ; </ms> </list> </apply> </list> </apply> </list> </list> </apply> <ms> - </ms> <apply> <ci> SuperscriptBox </ci> <ms> y </ms> <ms> 2 </ms> </apply> </list> </apply> <ms> , </ms> <apply> <ci> SuperscriptBox </ci> <ms> x </ms> <ms> 2 </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> </list> </apply> <ms> /; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> x </ms> <ms> ∈ </ms> <apply> <ci> TagBox </ci> <ms> ℝ </ms> <apply> <ci> Function </ci> <reals /> </apply> </apply> </list> </apply> <ms> ∧ </ms> <apply> <ci> RowBox </ci> <list> <ms> y </ms> <ms> ∈ </ms> <apply> <ci> TagBox </ci> <ms> ℝ </ms> <apply> <ci> Function </ci> <reals /> </apply> </apply> </list> </apply> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Im", "[", RowBox[List["EllipticE", "[", RowBox[List["x_", "+", RowBox[List["\[ImaginaryI]", " ", "y_"]]]], "]"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List["-", RowBox[List["(", RowBox[List["\[Pi]", " ", "y"]], ")"]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["3", "4"], ",", FractionBox["5", "4"], ",", FractionBox["1", "4"], ",", FractionBox["3", "4"]]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["1", ",", FractionBox["3", "2"]]], "}"]], ",", RowBox[List["{", FractionBox["3", "2"], "}"]], ",", RowBox[List["{", FractionBox["1", "2"], "}"]]]], "}"]], ",", RowBox[List["-", SuperscriptBox["y", "2"]]], ",", SuperscriptBox["x", "2"]]], "]"]]]], "-", RowBox[List[FractionBox["1", "64"], " ", RowBox[List["(", RowBox[List["3", " ", "\[Pi]", " ", "x", " ", "y"]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["5", "4"], ",", FractionBox["7", "4"], ",", FractionBox["3", "4"], ",", FractionBox["5", "4"]]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["2", ",", FractionBox["3", "2"]]], "}"]], ",", RowBox[List["{", FractionBox["3", "2"], "}"]], ",", RowBox[List["{", FractionBox["3", "2"], "}"]]]], "}"]], ",", RowBox[List["-", SuperscriptBox["y", "2"]]], ",", SuperscriptBox["x", "2"]]], "]"]]]]]], "/;", RowBox[List[RowBox[List["x", "\[Element]", "Reals"]], "&&", RowBox[List["y", "\[Element]", "Reals"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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