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http://functions.wolfram.com/08.03.06.0016.01
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EllipticPi[n, m] \[Proportional]
(Floor[Arg[x - n]/(2 Pi)] ((Pi I)/Sqrt[(x - 1) ((x - m)/x)])
(1 + O[n - x]))/E^(I Pi (Floor[(Pi - Arg[-m + n])/(2 Pi)] +
Floor[(Pi - Arg[(x - m)/x] - Arg[1 + (n - x)/(x - m)])/(2 Pi)])) +
EllipticPi[x, m] (1 + O[n - x]) /; (n -> x) && Element[x, Reals] && x > 1
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["EllipticPi", "[", RowBox[List["n", ",", "m"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List[RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["x", "-", "n"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["x", "-", "1"]], ")"]], FractionBox[RowBox[List["x", "-", "m"]], "x"]]]]], SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "-", RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "m"]], "+", "n"]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], "+", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "-", RowBox[List["Arg", "[", FractionBox[RowBox[List["x", "-", "m"]], "x"], "]"]], "-", RowBox[List["Arg", "[", RowBox[List["1", "+", FractionBox[RowBox[List["n", "-", "x"]], RowBox[List["x", "-", "m"]]]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], ")"]]]]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", RowBox[List["n", "-", "x"]], "]"]]]], ")"]]]], "+", RowBox[List[RowBox[List["EllipticPi", "[", RowBox[List["x", ",", "m"]], "]"]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", RowBox[List["n", "-", "x"]], "]"]]]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["n", "\[Rule]", "x"]], ")"]], "\[And]", RowBox[List["x", "\[Element]", "Reals"]], "\[And]", RowBox[List["x", ">", "1"]]]]]]]]
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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> Π </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ∝ </mo> <mrow> <mrow> <mrow> <mi> Π </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> x </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> O </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mi> x </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mtext> </mtext> <mrow> <mrow> <mo> ⌊ </mo> <mfrac> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> x </mi> <mo> - </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> <mo> ⌋ </mo> </mrow> <mo> ⁢ </mo> <mfrac> <mrow> <mtext> </mtext> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> </mrow> </mrow> <msqrt> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> x </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> x </mi> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mi> x </mi> </mfrac> </msqrt> </mfrac> <mo> ⁢ </mo> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> ⌊ </mo> <mfrac> <mrow> <mi> π </mi> <mo> - </mo> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> n </mi> <mo> - </mo> <mi> x </mi> </mrow> <mrow> <mi> x </mi> <mo> - </mo> <mi> m </mi> </mrow> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> x </mi> <mo> - </mo> <mi> m </mi> </mrow> <mi> x </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> <mo> ⌋ </mo> </mrow> <mo> + </mo> <mrow> <mo> ⌊ </mo> <mfrac> <mrow> <mi> π </mi> <mo> - </mo> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> <mo> ⌋ </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> O </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mi> x </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mi> x </mi> </mrow> <mo> ) </mo> </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> <mi> x </mi> <mo> > </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> EllipticPi </ci> <ci> n </ci> <ci> m </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <ci> EllipticPi </ci> <ci> x </ci> <ci> m </ci> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <ci> O </ci> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> x </ci> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <floor /> <apply> <times /> <apply> <arg /> <apply> <plus /> <ci> x </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <pi /> <imaginaryi /> <apply> <power /> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> x </ci> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <ci> x </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <apply> <power /> <ci> x </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <pi /> <apply> <plus /> <apply> <floor /> <apply> <times /> <apply> <plus /> <pi /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <arg /> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> x </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> x </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <arg /> <apply> <times /> <apply> <plus /> <ci> x </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <apply> <power /> <ci> x </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <floor /> <apply> <times /> <apply> <plus /> <pi /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <arg /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <ci> O </ci> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> x </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <ci> Rule </ci> <ci> n </ci> <ci> x </ci> </apply> <apply> <in /> <ci> x </ci> <reals /> </apply> <apply> <gt /> <ci> x </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EllipticPi", "[", RowBox[List["n_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["x", "-", "n"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], " ", RowBox[List["(", RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "-", RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "m"]], "+", "n"]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], "+", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "-", RowBox[List["Arg", "[", FractionBox[RowBox[List["x", "-", "m"]], "x"], "]"]], "-", RowBox[List["Arg", "[", RowBox[List["1", "+", FractionBox[RowBox[List["n", "-", "x"]], RowBox[List["x", "-", "m"]]]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], ")"]]]]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", RowBox[List["n", "-", "x"]], "]"]]]], ")"]]]], SqrtBox[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["x", "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["x", "-", "m"]], ")"]]]], "x"]]], "+", RowBox[List[RowBox[List["EllipticPi", "[", RowBox[List["x", ",", "m"]], "]"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", RowBox[List["n", "-", "x"]], "]"]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["n", "\[Rule]", "x"]], ")"]], "&&", RowBox[List["x", "\[Element]", "Reals"]], "&&", RowBox[List["x", ">", "1"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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