Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail CommentsSign the Guestbook

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

Download All Introductions For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











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/09.46.06.0017.01









  


  










Input Form





InverseJacobiSC[z, m] == InverseJacobiSC[z, Subscript[m, 0]] - ((I Sqrt[1 + z^2] Sqrt[1 + (1 - Subscript[m, 0]) z^2])/ (JacobiDC[InverseJacobiSC[z, Subscript[m, 0]], Subscript[m, 0]] JacobiNC[InverseJacobiSC[z, Subscript[m, 0]], Subscript[m, 0]])) Sum[((-1)^k/k!) Derivative[0, k][EllipticF][I ArcSinh[z], 1 - Subscript[m, 0]] (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[FractionBox[RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["1", "+", SuperscriptBox["z", "2"]]]], " ", SqrtBox[RowBox[List["1", "+", RowBox[List[RowBox[List["(", RowBox[List["1", "-", SubscriptBox["m", "0"]]], ")"]], " ", SuperscriptBox["z", "2"]]]]]]]], 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[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], RowBox[List["k", "!"]]], RowBox[List[SuperscriptBox["EllipticF", TagBox[RowBox[List["(", RowBox[List["0", ",", "k"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSinh", "[", "z", "]"]]]], ",", RowBox[List["1", "-", SubscriptBox["m", "0"]]]]], "]"]], 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> sc </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> &#63449; </mo> <mrow> <mrow> <msup> <mi> sc </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> z </mi> <mo> &#10072; </mo> <msub> <mi> m </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> m </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> <mrow> <mrow> <mi> dc </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <msup> <mi> sc </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> z </mi> <mo> &#10072; </mo> <msub> <mi> m </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> &#10072; </mo> <msub> <mi> m </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> nc </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <msup> <mi> sc </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> z </mi> <mo> &#10072; </mo> <msub> <mi> m </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> &#10072; </mo> <msub> <mi> m </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msup> <mi> EllipticF </mi> <semantics> <mrow> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;0&quot;, &quot;,&quot;, &quot;k&quot;]], &quot;)&quot;]], Derivative] </annotation> </semantics> </msup> <mo> ( </mo> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> sinh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> m </mi> <mn> 0 </mn> </msub> </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> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> InverseJacobiSC </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <plus /> <apply> <ci> InverseJacobiSC </ci> <ci> z </ci> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <imaginaryi /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <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 /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <ci> JacobiDC </ci> <apply> <ci> InverseJacobiSC </ci> <ci> z </ci> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <ci> JacobiNC </ci> <apply> <ci> InverseJacobiSC </ci> <ci> z </ci> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <power /> <apply> <factorial /> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <apply> <partialdiff /> <list> <cn type='integer'> 0 </cn> <ci> k </ci> </list> <ci> EllipticF </ci> </apply> <apply> <times /> <imaginaryi /> <apply> <arcsinh /> <ci> z </ci> </apply> </apply> <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> <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> </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["\[ImaginaryI]", " ", SqrtBox[RowBox[List["1", "+", SuperscriptBox["z", "2"]]]], " ", SqrtBox[RowBox[List["1", "+", RowBox[List[RowBox[List["(", RowBox[List["1", "-", SubscriptBox["mm", "0"]]], ")"]], " ", SuperscriptBox["z", "2"]]]]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List[SuperscriptBox["EllipticF", TagBox[RowBox[List["(", RowBox[List["0", ",", "k"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSinh", "[", "z", "]"]]]], ",", RowBox[List["1", "-", SubscriptBox["mm", "0"]]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", SubscriptBox["mm", "0"]]], ")"]], "k"]]], RowBox[List["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





© 1998-2014 Wolfram Research, Inc.