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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 at m==0





http://functions.wolfram.com/08.06.06.0087.01









  


  








Input Form





EllipticPi[n, z, m] \[Proportional] ((Pi/Sqrt[1 - n]) Round[Re[z]/Pi] + ArcTanh[Sqrt[-1 + n] Tan[z]]/ Sqrt[-1 + n]) (1 + O[m])








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["EllipticPi", "[", RowBox[List["n", ",", "z", ",", "m"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[FractionBox["\[Pi]", SqrtBox[RowBox[List["1", "-", "n"]]]], " ", RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]]]], "+", FractionBox[RowBox[List["ArcTanh", "[", RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", "n"]]], " ", RowBox[List["Tan", "[", "z", "]"]]]], "]"]], SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", "n"]]]]]], ")"]], RowBox[List["(", RowBox[List["1", " ", "+", RowBox[List["O", "[", "m", "]"]]]], ")"]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mi> &#928; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> ; </mo> <mrow> <mi> z </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8733; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <msup> <mi> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <msqrt> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <mi> tan </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <msqrt> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> </mfrac> <mo> + </mo> <mrow> <mfrac> <mi> &#960; </mi> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> n </mi> </mrow> </msqrt> </mfrac> <mo> &#8290; </mo> <mrow> <mo> &#8970; </mo> <mfrac> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mi> &#960; </mi> </mfrac> <mo> &#8969; </mo> </mrow> </mrow> </mrow> <mtext> </mtext> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> O </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <mi> &#928; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> ; </mo> <mrow> <mi> z </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8733; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <msup> <mi> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <msqrt> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <mi> tan </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <msqrt> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> </mfrac> <mo> + </mo> <mrow> <mfrac> <mi> &#960; </mi> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> n </mi> </mrow> </msqrt> </mfrac> <mo> &#8290; </mo> <mrow> <mo> &#8970; </mo> <mfrac> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mi> &#960; </mi> </mfrac> <mo> &#8969; </mo> </mrow> </mrow> </mrow> <mtext> </mtext> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> O </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EllipticPi", "[", RowBox[List["n_", ",", "z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["(", RowBox[List[FractionBox[RowBox[List["\[Pi]", " ", RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]]]], SqrtBox[RowBox[List["1", "-", "n"]]]], "+", FractionBox[RowBox[List["ArcTanh", "[", RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", "n"]]], " ", RowBox[List["Tan", "[", "z", "]"]]]], "]"]], SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", "n"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", "m", "]"]]]], ")"]]]]]]]]


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





2007-05-02