|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
http://functions.wolfram.com/08.06.04.0043.01
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Limit[EllipticPi[n, (3 Pi)/2 + 2 k Pi + I x - \[Epsilon], m],
\[Epsilon] -> Plus[0]] == -EllipticPi[n, (3 Pi)/2 + I x, m] +
4 (k + 2) EllipticPi[n, m] - (2/Sqrt[m]) EllipticPi[n/m, 1/m] /;
Element[m, Reals] && Element[x, Reals] &&
((0 < m < 1 && x > -Im[ArcCsc[Sqrt[m]]]) || (m > 1 && x < 0)) &&
Element[k, Integers]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Limit", "[", RowBox[List[RowBox[List["EllipticPi", "[", RowBox[List["n", ",", RowBox[List[FractionBox[RowBox[List["3", "\[Pi]"]], "2"], "+", RowBox[List["2", "k", " ", "\[Pi]"]], "+", RowBox[List["\[ImaginaryI]", " ", "x"]], "-", "\[Epsilon]"]], ",", "m"]], "]"]], ",", RowBox[List["\[Epsilon]", "\[Rule]", RowBox[List["+", "0"]]]]]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", RowBox[List["EllipticPi", "[", RowBox[List["n", ",", RowBox[List[FractionBox[RowBox[List["3", "\[Pi]"]], "2"], "+", RowBox[List["\[ImaginaryI]", " ", "x"]]]], ",", "m"]], "]"]]]], "+", RowBox[List["4", RowBox[List["(", RowBox[List["k", "+", "2"]], ")"]], " ", RowBox[List["EllipticPi", "[", RowBox[List["n", ",", "m"]], "]"]]]], "-", RowBox[List[FractionBox["2", SqrtBox["m"]], RowBox[List["EllipticPi", "[", RowBox[List[FractionBox["n", "m"], ",", FractionBox["1", "m"]]], "]"]]]]]]]], "/;", " ", RowBox[List[RowBox[List["m", "\[Element]", "Reals"]], "\[And]", RowBox[List["x", "\[Element]", "Reals"]], "\[And]", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["0", "<", "m", "<", "1"]], "\[And]", RowBox[List["x", ">", RowBox[List["-", RowBox[List["Im", "[", RowBox[List["ArcCsc", "[", RowBox[List["Sqrt", "[", "m", "]"]], "]"]], "]"]]]]]]]], ")"]], "\[Or]", RowBox[List["(", RowBox[List[RowBox[List["m", ">", "1"]], "\[And]", RowBox[List["x", "<", "0"]]]], ")"]]]], ")"]], "\[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> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> + </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> x </mi> </mrow> <mo> - </mo> <mi> ϵ </mi> <mo> + </mo> <mfrac> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> ❘ </mo> <mi> m </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <mn> 2 </mn> <mtext> </mtext> </mrow> <msqrt> <mi> m </mi> </msqrt> </mfrac> </mrow> <mo> ⁢ </mo> <mrow> <mi> Π </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mi> n </mi> <mi> m </mi> </mfrac> <mo> ❘ </mo> <mfrac> <mn> 1 </mn> <mi> m </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Π </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> Π </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> ; </mo> <mrow> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> x </mi> </mrow> <mo> + </mo> <mfrac> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> ❘ </mo> <mi> m </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <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> x </mi> <mo> ∈ </mo> <semantics> <mi> ℝ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalR]", Function[List[], Reals]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mn> 0 </mn> <mo> < </mo> <mi> m </mi> <mo> < </mo> <mn> 1 </mn> </mrow> <mo> ∧ </mo> <mrow> <mi> x </mi> <mo> > </mo> <mrow> <mo> - </mo> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msup> <mi> csc </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <msqrt> <mi> m </mi> </msqrt> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> ∨ </mo> <mrow> <mrow> <mi> m </mi> <mo> > </mo> <mn> 1 </mn> </mrow> <mo> ∧ </mo> <mrow> <mi> x </mi> <mo> < </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> <mo> ) </mo> </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 /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <ci> k </ci> </apply> <apply> <times /> <imaginaryi /> <ci> x </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ϵ </ci> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> m </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <power /> <ci> m </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> EllipticPi </ci> <apply> <times /> <ci> n </ci> <apply> <power /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <plus /> <ci> k </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> EllipticPi </ci> <ci> n </ci> <ci> m </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> EllipticPi </ci> <ci> n </ci> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> x </ci> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> m </ci> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> m </ci> <reals /> </apply> <apply> <in /> <ci> x </ci> <reals /> </apply> <apply> <or /> <apply> <and /> <apply> <lt /> <cn type='integer'> 0 </cn> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> <apply> <gt /> <ci> x </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <imaginary /> <apply> <arccsc /> <apply> <power /> <ci> m </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <gt /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> <apply> <lt /> <ci> x </ci> <cn type='integer'> 0 </cn> </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[FractionBox[RowBox[List["3", " ", "\[Pi]"]], "2"], "+", RowBox[List["2", " ", "k", " ", "\[Pi]"]], "+", RowBox[List["\[ImaginaryI]", " ", "x_"]], "-", "\[Epsilon]_"]], ",", "m_"]], "]"]], ",", RowBox[List["\[Epsilon]_", "\[Rule]", RowBox[List["+", "0"]]]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["EllipticPi", "[", RowBox[List["n", ",", RowBox[List[FractionBox[RowBox[List["3", " ", "\[Pi]"]], "2"], "+", RowBox[List["\[ImaginaryI]", " ", "x"]]]], ",", "m"]], "]"]]]], "+", RowBox[List["4", " ", RowBox[List["(", RowBox[List["k", "+", "2"]], ")"]], " ", RowBox[List["EllipticPi", "[", RowBox[List["n", ",", "m"]], "]"]]]], "-", FractionBox[RowBox[List["2", " ", RowBox[List["EllipticPi", "[", RowBox[List[FractionBox["n", "m"], ",", FractionBox["1", "m"]]], "]"]]]], SqrtBox["m"]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Reals"]], "&&", RowBox[List["x", "\[Element]", "Reals"]], "&&", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["0", "<", "m", "<", "1"]], "&&", RowBox[List["x", ">", RowBox[List["-", RowBox[List["Im", "[", RowBox[List["ArcCsc", "[", SqrtBox["m"], "]"]], "]"]]]]]]]], ")"]], "||", RowBox[List["(", RowBox[List[RowBox[List["m", ">", "1"]], "&&", RowBox[List["x", "<", "0"]]]], ")"]]]], ")"]], "&&", RowBox[List["k", "\[Element]", "Integers"]]]]]]]]]] |
|
|
|
|
|
|
|
|
|
|
|
Date Added to functions.wolfram.com (modification date)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
){kind=small}
