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

 

Developed with Mathematica -- Download a Free Trial Version
 











RamanujanTauTheta






Mathematica Notation

Traditional Notation









Zeta Functions and Polylogarithms > RamanujanTauTheta[z] > Series representations > Generalized power series > Expansions on branch cuts > For the function itself > In the upper half-plane





http://functions.wolfram.com/10.12.06.0007.01









  


  










Input Form





RamanujanTauTheta[z] == RamanujanTauTheta[Subscript[z, 0]] + Pi (6 + Floor[I Subscript[z, 0]]) Floor[Arg[I (z - Subscript[z, 0])]/ (2 Pi)] + (I/2) Sum[(I^k/k!) ((-1)^k PolyGamma[-1 + k, 6 - I Subscript[z, 0]] - PolyGamma[-1 + k, 6 + I Subscript[z, 0]] + 2 Log[2 Pi] KroneckerDelta[k - 1]) (z - Subscript[z, 0])^k, {k, 1, Infinity}] /; Element[I Subscript[z, 0], Reals] && I Subscript[z, 0] < -6 && !Element[I Subscript[z, 0], Integers]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["RamanujanTauTheta", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List["RamanujanTauTheta", "[", SubscriptBox["z", "0"], "]"]], "+", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["6", "+", RowBox[List["Floor", "[", RowBox[List["\[ImaginaryI]", " ", SubscriptBox["z", "0"]]], "]"]]]], ")"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "+", RowBox[List[FractionBox["\[ImaginaryI]", "2"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List[FractionBox[SuperscriptBox["\[ImaginaryI]", "k"], RowBox[List["k", "!"]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "k"]], ",", RowBox[List["6", "-", RowBox[List["\[ImaginaryI]", " ", SubscriptBox["z", "0"]]]]]]], "]"]]]], "-", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "k"]], ",", RowBox[List["6", "+", RowBox[List["\[ImaginaryI]", " ", SubscriptBox["z", "0"]]]]]]], "]"]], "+", RowBox[List["2", " ", RowBox[List["Log", "[", RowBox[List["2", " ", "\[Pi]"]], "]"]], RowBox[List["KroneckerDelta", "[", RowBox[List["k", "-", "1"]], "]"]]]]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]], "k"]]]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["\[ImaginaryI]", " ", SubscriptBox["z", "0"]]], "\[Element]", "Reals"]], "\[And]", RowBox[List[RowBox[List["\[ImaginaryI]", " ", SubscriptBox["z", "0"]]], "<", RowBox[List["-", "6"]]]], "\[And]", RowBox[List["Not", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", SubscriptBox["z", "0"]]], "\[Element]", "Integers"]], "]"]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mi> &#964;&#952; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mrow> <mrow> <mi> &#964;&#952; </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mo> &#8970; </mo> <mfrac> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mfrac> <mo> &#8971; </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 6 </mn> <mo> + </mo> <mrow> <mo> &#8970; </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> &#8971; </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mfrac> <mrow> <msup> <mi> &#8520; </mi> <mi> k </mi> </msup> <mtext> </mtext> </mrow> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mn> 6 </mn> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mo> &#8290; </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mn> 6 </mn> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[List[], Reals]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> &lt; </mo> <mrow> <mo> - </mo> <mn> 6 </mn> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> &#8713; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[List[], Integers]] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> &#964;&#952; </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <ci> &#964;&#952; </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <times /> <pi /> <apply> <floor /> <apply> <times /> <apply> <arg /> <apply> <times /> <imaginaryi /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 6 </cn> <apply> <floor /> <apply> <times /> <imaginaryi /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <imaginaryi /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <imaginaryi /> <ci> k </ci> </apply> <apply> <power /> <apply> <factorial /> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> KroneckerDelta </ci> <apply> <plus /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ln /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <cn type='integer'> 6 </cn> <apply> <times /> <imaginaryi /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <cn type='integer'> 6 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <apply> <times /> <imaginaryi /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <reals /> </apply> <apply> <lt /> <apply> <times /> <imaginaryi /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> -6 </cn> </apply> <apply> <notin /> <apply> <times /> <imaginaryi /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["RamanujanTauTheta", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["RamanujanTauTheta", "[", SubscriptBox["zz", "0"], "]"]], "+", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["6", "+", RowBox[List["Floor", "[", RowBox[List["\[ImaginaryI]", " ", SubscriptBox["zz", "0"]]], "]"]]]], ")"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox["\[ImaginaryI]", "k"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "k"]], ",", RowBox[List["6", "-", RowBox[List["\[ImaginaryI]", " ", SubscriptBox["zz", "0"]]]]]]], "]"]]]], "-", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "k"]], ",", RowBox[List["6", "+", RowBox[List["\[ImaginaryI]", " ", SubscriptBox["zz", "0"]]]]]]], "]"]], "+", RowBox[List["2", " ", RowBox[List["Log", "[", RowBox[List["2", " ", "\[Pi]"]], "]"]], " ", RowBox[List["KroneckerDelta", "[", RowBox[List["k", "-", "1"]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]], "k"]]], RowBox[List["k", "!"]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["\[ImaginaryI]", " ", SubscriptBox["zz", "0"]]], "\[Element]", "Reals"]], "&&", RowBox[List[RowBox[List["\[ImaginaryI]", " ", SubscriptBox["zz", "0"]]], "<", RowBox[List["-", "6"]]]], "&&", RowBox[List["!", RowBox[List[RowBox[List["\[ImaginaryI]", " ", SubscriptBox["zz", "0"]]], "\[Element]", "Integers"]]]]]]]]]]]]










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





2007-05-02





© 1998-2014 Wolfram Research, Inc.