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variants of this functions
EllipticPi






Mathematica Notation

Traditional Notation









Elliptic Integrals > EllipticPi[n,z,m] > Series representations > Generalized power series > Expansions on branch cuts > Formulas on real axis for real m, n > For n<1<m,csc-1(m1/2)+Pi u<xu+1/2)/;uZ





http://functions.wolfram.com/08.06.06.0034.01









  


  










Input Form





EllipticPi[n, z, m] \[Proportional] (Exp[(-Pi) I Floor[Arg[x - z]/(2 Pi)]] EllipticPi[n, x, m] + ((1/Sqrt[m]) EllipticPi[n/m, 1/m] + 2 (Floor[x/Pi - 1/2] + 1) EllipticPi[n, m]) (1 - Exp[(-Pi) I Floor[Arg[x - z]/(2 Pi)]])) (1 + O[z - x]) /; (z -> x) && Element[x, Reals] && Element[m, Reals] && Element[n, Reals] && n < 1 < m && ArcCsc[Sqrt[m]] + Pi u < x < Pi/2 + Pi u && Element[u, Integers]










Standard Form





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MathML Form







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</mo> <mrow> <mi> m </mi> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[List[], Reals]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[List[], Reals]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> n </mi> <mo> &lt; </mo> <mn> 1 </mn> <mo> &lt; </mo> <mi> m </mi> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> u </mi> </mrow> <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> </mrow> <mo> &lt; </mo> <mi> x </mi> <mo> &lt; </mo> <mrow> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> u </mi> </mrow> <mo> + </mo> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> &#8743; 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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EllipticPi", "[", RowBox[List["n_", ",", "z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[Pi]"]], " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["x", "-", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List["EllipticPi", "[", RowBox[List["n", ",", "x", ",", "m"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[FractionBox[RowBox[List["EllipticPi", "[", RowBox[List[FractionBox["n", "m"], ",", FractionBox["1", "m"]]], "]"]], SqrtBox["m"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["Floor", "[", RowBox[List[FractionBox["x", "\[Pi]"], "-", FractionBox["1", "2"]]], "]"]], "+", "1"]], ")"]], " ", RowBox[List["EllipticPi", "[", RowBox[List["n", ",", "m"]], "]"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[Pi]"]], " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["x", "-", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]]]], ")"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", RowBox[List["z", "-", "x"]], "]"]]]], ")"]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["z", "\[Rule]", "x"]], ")"]], "&&", RowBox[List["x", "\[Element]", "Reals"]], "&&", RowBox[List["m", "\[Element]", "Reals"]], "&&", RowBox[List["n", "\[Element]", "Reals"]], "&&", RowBox[List["n", "<", "1", "<", "m"]], "&&", RowBox[List[RowBox[List[RowBox[List["ArcCsc", "[", SqrtBox["m"], "]"]], "+", RowBox[List["\[Pi]", " ", "u"]]]], "<", "x", "<", RowBox[List[FractionBox["\[Pi]", "2"], "+", RowBox[List["\[Pi]", " ", "u"]]]]]], "&&", RowBox[List["u", "\[Element]", "Integers"]]]]]]]]]]










Date Added to functions.wolfram.com (modification date)





2007-05-02





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