Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











KleinInvariantJ






Mathematica Notation

Traditional Notation









Elliptic Functions > KleinInvariantJ[z] > Differentiation > Low-order differentiation





http://functions.wolfram.com/09.50.20.0003.01









  


  










Input Form





D[KleinInvariantJ[z], z] == ((I Pi)/864) (-E^(-2 I Pi z) + Sum[k Subscript[a, k] E^(2 k I Pi z), {k, 1, Infinity}]) /; Subscript[a, k] == ((2 Pi)/Sqrt[k]) Sum[(Subscript[A, j][k]/j) BesselI[1, ((4 Pi)/j) Sqrt[k]], {j, 1, Infinity}] && (Subscript[A, j][k] == Sum[KroneckerDelta[1, GCD[h, j]] Exp[(-((2 Pi I)/j)) (h k + H[j, h])], {h, 0, j - 1}] /; Mod[h H[j, h], j] == -1)










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", "z"], RowBox[List["KleinInvariantJ", "[", "z", "]"]]]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "864"], RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "z"]]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List["k", " ", SubscriptBox["a", "k"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "k", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "z"]]]]]]]]], ")"]]]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List[SubscriptBox["a", "k"], "\[Equal]", RowBox[List[FractionBox[RowBox[List[RowBox[List["2", " ", "\[Pi]"]], " "]], SqrtBox["k"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[RowBox[List[SubscriptBox["A", "j"], "[", "k", "]"]], " "]], RowBox[List["j", " "]]], RowBox[List["BesselI", "[", RowBox[List["1", ",", RowBox[List[FractionBox[RowBox[List["4", " ", "\[Pi]"]], "j"], " ", SqrtBox["k"]]]]], "]"]]]]]]]]]], "\[And]", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["A", "j"], "[", "k", "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "0"]], RowBox[List["j", "-", "1"]]], RowBox[List[RowBox[List["KroneckerDelta", "[", RowBox[List["1", ",", " ", RowBox[List["GCD", "[", RowBox[List["h", ",", "j"]], "]"]]]], "]"]], RowBox[List["Exp", "[", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]"]], " "]], "j"]]], RowBox[List["(", RowBox[List[RowBox[List["h", " ", "k"]], "+", RowBox[List["H", "[", RowBox[List["j", ",", "h"]], "]"]]]], ")"]]]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List["Mod", "[", RowBox[List[RowBox[List["h", " ", RowBox[List["H", "[", RowBox[List["j", ",", "h"]], "]"]]]], ",", "j"]], "]"]], "\[Equal]", RowBox[List["-", "1"]]]]]], ")"]]]], ")"]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mfrac> <mrow> <mo> &#8706; </mo> <semantics> <mrow> <mi> J </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;J&quot;, &quot;(&quot;, TagBox[&quot;z&quot;, Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[KleinInvariantJ[Slot[1]]]]] </annotation> </semantics> </mrow> <mrow> <mo> &#8706; </mo> <mi> z </mi> </mrow> </mfrac> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> <mn> 864 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> </mrow> <mo> + </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mi> k </mi> <mo> &#8290; </mo> <msub> <mi> a </mi> <mi> k </mi> </msub> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <msub> <mi> a </mi> <mi> k </mi> </msub> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> <mtext> </mtext> </mrow> <msqrt> <mi> k </mi> </msqrt> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mfrac> <mn> 1 </mn> <mi> j </mi> </mfrac> <mo> &#8290; </mo> <mrow> <msub> <mi> A </mi> <mi> j </mi> </msub> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> I </mi> <mn> 1 </mn> </msub> <mo> ( </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msqrt> <mi> k </mi> </msqrt> </mrow> <mi> j </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <msub> <mi> A </mi> <mi> j </mi> </msub> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> h </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> j </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mn> 1 </mn> <mo> , </mo> <mrow> <mi> gcd </mi> <mo> [ </mo> <mrow> <mi> h </mi> <mo> , </mo> <mi> j </mi> </mrow> <mo> ] </mo> </mrow> </mrow> </msub> <mo> &#8290; </mo> <mrow> <mi> exp </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> h </mi> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mrow> <mi> H </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> j </mi> <mo> , </mo> <mi> h </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mi> j </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <semantics> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> h </mi> <mo> &#8290; </mo> <mrow> <mi> H </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> j </mi> <mo> , </mo> <mi> h </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> mod </mi> <mo> &#8290; </mo> <mi> j </mi> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <rem /> <apply> <times /> <ci> FE`Conversion`Private`h </ci> <apply> <ci> FE`Conversion`Private`H </ci> <ci> $CellContext`j </ci> <ci> FE`Conversion`Private`h </ci> </apply> </apply> <ci> $CellContext`j </ci> </apply> </annotation-xml> </semantics> <mo> &#10869; </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> z </ci> </bvar> <apply> <ci> KleinInvariantJ </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <times /> <pi /> <imaginaryi /> <apply> <power /> <cn type='integer'> 864 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -2 </cn> <imaginaryi /> <pi /> <ci> z </ci> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <ci> k </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <ci> k </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> <imaginaryi /> <pi /> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> a </ci> <ci> k </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <apply> <power /> <apply> <power /> <ci> k </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> j </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <apply> <ci> Subscript </ci> <ci> A </ci> <ci> j </ci> </apply> <ci> k </ci> </apply> <apply> <ci> BesselI </ci> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <times /> <cn type='integer'> 4 </cn> <pi /> </apply> <apply> <power /> <ci> k </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> j </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <apply> <ci> Subscript </ci> <ci> A </ci> <ci> j </ci> </apply> <ci> k </ci> </apply> <apply> <sum /> <bvar> <ci> h </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> j </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <times /> <apply> <ci> KroneckerDelta </ci> <cn type='integer'> 1 </cn> <apply> <ci> gcd </ci> <ci> h </ci> <ci> j </ci> </apply> </apply> <apply> <exp /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <imaginaryi /> </apply> <apply> <plus /> <apply> <times /> <ci> h </ci> <ci> k </ci> </apply> <apply> <ci> H </ci> <ci> j </ci> <ci> h </ci> </apply> </apply> <apply> <power /> <ci> j </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <eq /> <apply> <rem /> <apply> <times /> <ci> FE`Conversion`Private`h </ci> <apply> <ci> FE`Conversion`Private`H </ci> <ci> $CellContext`j </ci> <ci> FE`Conversion`Private`h </ci> </apply> </apply> <ci> $CellContext`j </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["z_"]]], RowBox[List["KleinInvariantJ", "[", "z_", "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox["1", "864"], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "z"]]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List["k", " ", SubscriptBox["a", "k"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "k", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "z"]]]]]]]]], ")"]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["a", "k"], "\[Equal]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", "\[Pi]"]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List[SubscriptBox["A", "j"], "[", "k", "]"]], " ", RowBox[List["BesselI", "[", RowBox[List["1", ",", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["4", " ", "\[Pi]"]], ")"]], " ", SqrtBox["k"]]], "j"]]], "]"]]]], "j"]]]]], SqrtBox["k"]]]], "&&", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["A", "j"], "[", "k", "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "0"]], RowBox[List["j", "-", "1"]]], RowBox[List[RowBox[List["KroneckerDelta", "[", RowBox[List["1", ",", RowBox[List["GCD", "[", RowBox[List["h", ",", "j"]], "]"]]]], "]"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["h", " ", "k"]], "+", RowBox[List["H", "[", RowBox[List["j", ",", "h"]], "]"]]]], ")"]]]], "j"]]]]]]]]]], "/;", RowBox[List[RowBox[List["Mod", "[", RowBox[List[RowBox[List["h", " ", RowBox[List["H", "[", RowBox[List["j", ",", "h"]], "]"]]]], ",", "j"]], "]"]], "\[Equal]", RowBox[List["-", "1"]]]]]], ")"]]]]]]]]]]










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





2001-10-29