|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
http://functions.wolfram.com/08.06.04.0036.01
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Limit[EllipticPi[n, x - I \[Epsilon], m], \[Epsilon] -> Plus[0]] ==
EllipticPi[n, x, m] /; Element[x, Reals] && Element[m, Reals] &&
Element[n, Reals] && 1 < m < n && ArcCsc[Sqrt[n]] + Pi k < x <
Pi k + Pi/2 && Element[k, Integers]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Limit", "[", RowBox[List[RowBox[List["EllipticPi", "[", RowBox[List["n", ",", RowBox[List["x", "-", RowBox[List["\[ImaginaryI]", " ", "\[Epsilon]"]]]], ",", "m"]], "]"]], ",", RowBox[List["\[Epsilon]", "\[Rule]", RowBox[List["+", "0"]]]]]], "]"]], "\[Equal]", RowBox[List["EllipticPi", "[", RowBox[List["n", ",", "x", ",", "m"]], "]"]]]], "/;", " ", RowBox[List[RowBox[List["x", "\[Element]", "Reals"]], "\[And]", RowBox[List["m", "\[Element]", "Reals"]], "\[And]", RowBox[List["n", "\[Element]", "Reals"]], "\[And]", RowBox[List["1", "<", "m", "<", "n"]], "\[And]", RowBox[List[RowBox[List[RowBox[List["ArcCsc", "[", SqrtBox["n"], "]"]], "+", RowBox[List["\[Pi]", " ", "k"]]]], "<", "x", "<", RowBox[List[RowBox[List["\[Pi]", " ", "k"]], "+", FractionBox["\[Pi]", "2"]]]]], "\[And]", RowBox[List["k", "\[Element]", "Integers"]]]]]]]]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <munder> <mi> lim </mi> <mrow> <mi> ϵ </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mrow> <mo> + </mo> <mn> 0 </mn> </mrow> </mrow> </munder> <mo> ⁢ </mo> <mtext>   </mtext> <mrow> <mi> Π </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> ; </mo> <mrow> <mrow> <mi> x </mi> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> ϵ </mi> </mrow> </mrow> <mo> ❘ </mo> <mi> m </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mi> Π </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> ; </mo> <mrow> <mi> x </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> x </mi> <mo> ∈ </mo> <semantics> <mi> ℝ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalR]", Function[List[], Reals]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> m </mi> <mo> ∈ </mo> <semantics> <mi> ℝ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalR]", Function[List[], Reals]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <semantics> <mi> ℝ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalR]", Function[List[], Reals]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mn> 1 </mn> <mo> < </mo> <mi> m </mi> <mo> < </mo> <mi> n </mi> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> + </mo> <mrow> <msup> <mi> csc </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <msqrt> <mi> n </mi> </msqrt> <mo> ) </mo> </mrow> </mrow> <mo> < </mo> <mi> x </mi> <mo> < </mo> <mrow> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> + </mo> <mfrac> <mi> π </mi> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mi> k </mi> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <limit /> <bvar> <ci> ϵ </ci> </bvar> <condition> <apply> <tendsto /> <ci> ϵ </ci> <apply> <plus /> <cn type='integer'> 0 </cn> </apply> </apply> </condition> <apply> <ci> EllipticPi </ci> <ci> n </ci> <apply> <plus /> <ci> x </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> ϵ </ci> </apply> </apply> </apply> <ci> m </ci> </apply> </apply> <apply> <ci> EllipticPi </ci> <ci> n </ci> <ci> x </ci> <ci> m </ci> </apply> </apply> <apply> <and /> <apply> <in /> <ci> x </ci> <reals /> </apply> <apply> <in /> <ci> m </ci> <reals /> </apply> <apply> <in /> <ci> n </ci> <reals /> </apply> <apply> <lt /> <cn type='integer'> 1 </cn> <ci> m </ci> <ci> n </ci> </apply> <apply> <lt /> <apply> <plus /> <apply> <times /> <pi /> <ci> k </ci> </apply> <apply> <arccsc /> <apply> <power /> <ci> n </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <ci> x </ci> <apply> <plus /> <apply> <times /> <pi /> <ci> k </ci> </apply> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <in /> <ci> k </ci> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
|
|
|
|
|
|
|
|
|
|
| |
|
|
|
|
| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Limit", "[", RowBox[List[RowBox[List["EllipticPi", "[", RowBox[List["n_", ",", RowBox[List["x_", "-", RowBox[List["\[ImaginaryI]", " ", "\[Epsilon]_"]]]], ",", "m_"]], "]"]], ",", RowBox[List["\[Epsilon]_", "\[Rule]", RowBox[List["+", "0"]]]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["EllipticPi", "[", RowBox[List["n", ",", "x", ",", "m"]], "]"]], "/;", RowBox[List[RowBox[List["x", "\[Element]", "Reals"]], "&&", RowBox[List["m", "\[Element]", "Reals"]], "&&", RowBox[List["n", "\[Element]", "Reals"]], "&&", RowBox[List["1", "<", "m", "<", "n"]], "&&", RowBox[List[RowBox[List[RowBox[List["ArcCsc", "[", SqrtBox["n"], "]"]], "+", RowBox[List["\[Pi]", " ", "k"]]]], "<", "x", "<", RowBox[List[RowBox[List["\[Pi]", " ", "k"]], "+", FractionBox["\[Pi]", "2"]]]]], "&&", RowBox[List["k", "\[Element]", "Integers"]]]]]]]]]] |
|
|
|
|
|
|
|
|
|
|
Date Added to functions.wolfram.com (modification date)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|