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http://functions.wolfram.com/08.04.04.0007.01
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BranchPoints[EllipticE[z, m], z] ==
{SequenceList[ArcSin[1/Sqrt[m]] + Pi k, Element[k, Integers]],
SequenceList[-ArcSin[1/Sqrt[m]] + Pi k, Element[k, Integers]],
SequenceList[Pi/2 + Pi k, Element[k, Integers] &&
!IntervalMemberQ[Interval[{0, 1}], m]], ComplexInfinity}
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Cell[BoxData[RowBox[List[RowBox[List["BranchPoints", "[", RowBox[List[RowBox[List["EllipticE", "[", RowBox[List["z", ",", "m"]], "]"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List["{", RowBox[List[RowBox[List["SequenceList", "[", RowBox[List[RowBox[List[RowBox[List["ArcSin", "[", FractionBox["1", SqrtBox["m"]], "]"]], "+", RowBox[List["\[Pi]", " ", "k"]]]], ",", RowBox[List["k", "\[Element]", "Integers"]]]], "]"]], ",", RowBox[List["SequenceList", "[", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["ArcSin", "[", FractionBox["1", SqrtBox["m"]], "]"]]]], "+", RowBox[List["\[Pi]", " ", "k"]]]], ",", RowBox[List["k", "\[Element]", "Integers"]]]], "]"]], ",", RowBox[List["SequenceList", "[", RowBox[List[RowBox[List[FractionBox["\[Pi]", "2"], "+", RowBox[List["\[Pi]", " ", "k"]]]], ",", RowBox[List[RowBox[List["k", "\[Element]", "Integers"]], "\[And]", RowBox[List["Not", "[", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List["0", ",", "1"]], "}"]], "]"]], ",", "m"]], "]"]], "]"]]]]]], "]"]], ",", "ComplexInfinity"]], "}"]]]]]]
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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> <msub> <mi> ℬ𝒫 </mi> <mi> a </mi> </msub> <mo> ( </mo> <semantics> <mrow> <mi> E </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["E", "(", RowBox[List["z", "\[VerticalSeparator]", "m"]], ")"]], InterpretTemplate[Function[List[$CellContext`a, $CellContext`b], Zeta[$CellContext`a, $CellContext`b]]]] </annotation> </semantics> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mo> { </mo> <mrow> <mrow> <mo> { </mo> <mrow> <mrow> <mrow> <msup> <mi> sin </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mn> 1 </mn> <msqrt> <mi> m </mi> </msqrt> </mfrac> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> k </mi> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> k </mi> <mo> ∈ </mo> <mi> ℤ </mi> </mrow> </mrow> <mo> } </mo> </mrow> <mo> , </mo> <mrow> <mo> { </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mrow> <msup> <mi> sin </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mn> 1 </mn> <msqrt> <mi> m </mi> </msqrt> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> k </mi> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> k </mi> <mo> ∈ </mo> <mi> ℤ </mi> </mrow> </mrow> <mo> } </mo> </mrow> <mo> , </mo> <mrow> <mo> { </mo> <mrow> <mrow> <mfrac> <mi> π </mi> <mn> 2 </mn> </mfrac> <mo> + </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> k </mi> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> k </mi> <mo> ∈ </mo> <mi> ℤ </mi> </mrow> <mo> ∧ </mo> <mrow> <mi> m </mi> <mo> ∉ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> } </mo> </mrow> <mo> , </mo> <mover> <mi> ∞ </mi> <mo> ~ </mo> </mover> </mrow> <mo> } </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <msub> <mi> ℬ𝒫 </mi> <mi> a </mi> </msub> <mo> ( </mo> <semantics> <mrow> <mi> E </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["E", "(", RowBox[List["z", "\[VerticalSeparator]", "m"]], ")"]], InterpretTemplate[Function[List[$CellContext`a, $CellContext`b], Zeta[$CellContext`a, $CellContext`b]]]] </annotation> </semantics> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mo> { </mo> <mrow> <mrow> <mo> { </mo> <mrow> <mrow> <mrow> <msup> <mi> sin </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mn> 1 </mn> <msqrt> <mi> m </mi> </msqrt> </mfrac> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> k </mi> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> k </mi> <mo> ∈ </mo> <mi> ℤ </mi> </mrow> </mrow> <mo> } </mo> </mrow> <mo> , </mo> <mrow> <mo> { </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mrow> <msup> <mi> sin </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mn> 1 </mn> <msqrt> <mi> m </mi> </msqrt> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> k </mi> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> k </mi> <mo> ∈ </mo> <mi> ℤ </mi> </mrow> </mrow> <mo> } </mo> </mrow> <mo> , </mo> <mrow> <mo> { </mo> <mrow> <mrow> <mfrac> <mi> π </mi> <mn> 2 </mn> </mfrac> <mo> + </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> k </mi> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> k </mi> <mo> ∈ </mo> <mi> ℤ </mi> </mrow> <mo> ∧ </mo> <mrow> <mi> m </mi> <mo> ∉ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> } </mo> </mrow> <mo> , </mo> <mover> <mi> ∞ </mi> <mo> ~ </mo> </mover> </mrow> <mo> } </mo> </mrow> </mrow> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["BranchPoints", "[", RowBox[List[RowBox[List["EllipticE", "[", RowBox[List["z_", ",", "m_"]], "]"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List["{", RowBox[List[RowBox[List["SequenceList", "[", RowBox[List[RowBox[List[RowBox[List["ArcSin", "[", FractionBox["1", SqrtBox["m"]], "]"]], "+", RowBox[List["\[Pi]", " ", "k"]]]], ",", RowBox[List["k", "\[Element]", "Integers"]]]], "]"]], ",", RowBox[List["SequenceList", "[", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["ArcSin", "[", FractionBox["1", SqrtBox["m"]], "]"]]]], "+", RowBox[List["\[Pi]", " ", "k"]]]], ",", RowBox[List["k", "\[Element]", "Integers"]]]], "]"]], ",", RowBox[List["SequenceList", "[", RowBox[List[RowBox[List[FractionBox["\[Pi]", "2"], "+", RowBox[List["\[Pi]", " ", "k"]]]], ",", RowBox[List[RowBox[List["k", "\[Element]", "Integers"]], "&&", RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List["0", ",", "1"]], "}"]], "]"]], ",", "m"]], "]"]]]]]]]], "]"]], ",", "ComplexInfinity"]], "}"]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
