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InverseJacobiSC






Mathematica Notation

Traditional Notation









Elliptic Functions > InverseJacobiSC[z,m] > Series representations > Generalized power series > Expansions at generic point m==m0





http://functions.wolfram.com/09.46.06.0016.01









  


  










Input Form





InverseJacobiSC[z, m] == InverseJacobiSC[z, Subscript[m, 0]] + ((Sqrt[z^2 + 1] Sqrt[(1 - Subscript[m, 0]) z^2 + 1])/ (JacobiDC[InverseJacobiSC[z, Subscript[m, 0]], Subscript[m, 0]] JacobiNC[InverseJacobiSC[z, Subscript[m, 0]], Subscript[m, 0]])) Sum[((Pochhammer[1/2, k] z^(1 + 2 k))/(k! (1 + 2 k))) AppellF1[1/2 + k, 1/2, 1/2 + k, 3/2 + k, -z^2, (-(1 - Subscript[m, 0])) z^2] (m - Subscript[m, 0])^k, {k, 1, Infinity}]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["InverseJacobiSC", "[", RowBox[List["z", ",", "m"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["InverseJacobiSC", "[", RowBox[List["z", ",", SubscriptBox["m", "0"]]], "]"]], "+", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "+", "1"]]], SqrtBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "-", SubscriptBox["m", "0"]]], ")"]], " ", SuperscriptBox["z", "2"]]], "+", "1"]]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["JacobiDC", "[", RowBox[List[RowBox[List["InverseJacobiSC", "[", RowBox[List["z", ",", SubscriptBox["m", "0"]]], "]"]], ",", SubscriptBox["m", "0"]]], "]"]], " ", RowBox[List["JacobiNC", "[", RowBox[List[RowBox[List["InverseJacobiSC", "[", RowBox[List["z", ",", SubscriptBox["m", "0"]]], "]"]], ",", SubscriptBox["m", "0"]]], "]"]]]], ")"]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], SuperscriptBox["z", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]]]]], RowBox[List[RowBox[List["k", "!"]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]], ")"]]]]], " ", RowBox[List["AppellF1", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "k"]], ",", FractionBox["1", "2"], ",", RowBox[List[FractionBox["1", "2"], "+", "k"]], ",", RowBox[List[FractionBox["3", "2"], "+", "k"]], ",", RowBox[List["-", SuperscriptBox["z", "2"]]], ",", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["1", "-", SubscriptBox["m", "0"]]], ")"]]]], " ", SuperscriptBox["z", "2"]]]]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", SubscriptBox["m", "0"]]], ")"]], "k"]]]]]]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msup> <mi> cs </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> z </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mo> &#8290; </mo> <msqrt> <mi> &#960; </mi> </msqrt> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> </mrow> <mrow> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <semantics> <msub> <mi> F </mi> <mn> 1 </mn> </msub> <annotation-xml encoding='MathML-Content'> <ci> AppellF1 </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> k </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ; </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mrow> <mi> k </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ; </mo> <mrow> <mi> k </mi> <mo> + </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ; </mo> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> m </mi> <mn> 0 </mn> </msub> </mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <msub> <mi> m </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> InverseJacobiCS </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <ci> k </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> AppellF1 </ci> <apply> <plus /> <ci> k </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <ci> k </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <ci> k </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <ci> k </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["InverseJacobiSC", "[", RowBox[List["z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["InverseJacobiSC", "[", RowBox[List["z", ",", SubscriptBox["mm", "0"]]], "]"]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "+", "1"]]], " ", SqrtBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "-", SubscriptBox["mm", "0"]]], ")"]], " ", SuperscriptBox["z", "2"]]], "+", "1"]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], " ", SuperscriptBox["z", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]]]]], ")"]], " ", RowBox[List["AppellF1", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "k"]], ",", FractionBox["1", "2"], ",", RowBox[List[FractionBox["1", "2"], "+", "k"]], ",", RowBox[List[FractionBox["3", "2"], "+", "k"]], ",", RowBox[List["-", SuperscriptBox["z", "2"]]], ",", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["1", "-", SubscriptBox["mm", "0"]]], ")"]]]], " ", SuperscriptBox["z", "2"]]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", SubscriptBox["mm", "0"]]], ")"]], "k"]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]], ")"]]]]]]]]], RowBox[List[RowBox[List["JacobiDC", "[", RowBox[List[RowBox[List["InverseJacobiSC", "[", RowBox[List["z", ",", SubscriptBox["mm", "0"]]], "]"]], ",", SubscriptBox["mm", "0"]]], "]"]], " ", RowBox[List["JacobiNC", "[", RowBox[List[RowBox[List["InverseJacobiSC", "[", RowBox[List["z", ",", SubscriptBox["mm", "0"]]], "]"]], ",", SubscriptBox["mm", "0"]]], "]"]]]]]]]]]]]










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





2007-05-02





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