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

Download All Introductions For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











variants of this functions
EllipticPi






Mathematica Notation

Traditional Notation









Elliptic Integrals > EllipticPi[n,z,m] > Series representations > Generalized power series > Expansions on branch cuts > Formulas for vertical intervals > For Re(z0/2 Pi-3/4) ∈ Z





http://functions.wolfram.com/08.06.06.0066.01









  


  










Input Form





EllipticPi[n, z, m] \[Proportional] (2 (2 Re[Subscript[z, 0]/(2 Pi) - 3/4] + 1) EllipticPi[n, m] + (1/Sqrt[m]) EllipticPi[n/m, 1/m]) (1 - Exp[(-Pi) I (Floor[1/4 + Arg[z - Subscript[z, 0]]/(2 Pi)] + Floor[1/4 - Arg[z - Subscript[z, 0]]/(2 Pi)])]) + Exp[(-Pi) I (Floor[1/4 + Arg[z - Subscript[z, 0]]/(2 Pi)] + Floor[1/4 - Arg[z - Subscript[z, 0]]/(2 Pi)])] EllipticPi[n, Subscript[z, 0], m] + Sum[(1/p!) Sum[(1/j!) Sum[Binomial[j, Subscript[k, 1]] Sum[((-1)^Subscript[k, 1] 2^(Subscript[k, 1] - j) Sin[Subscript[z, 0]]^Subscript[k, 1] (2 Subscript[k, 2] + Subscript[k, 1] - j)^p Binomial[j - Subscript[k, 1], Subscript[k, 2]] Sum[(Pochhammer[1 - j, 2 (j - i) - 2]/( (j - i - 1)! (2 Sin[Subscript[z, 0]])^(j - 2 i - 1))) Sum[(-1)^Subscript[i, 1] KroneckerDelta[Subscript[i, 1] + Subscript[i, 2] + Subscript[i, 3] - i] Multinomial[ Subscript[i, 1], Subscript[i, 2], Subscript[i, 3]] n^Subscript[i, 1] m^Subscript[i, 3] Pochhammer[ -Subscript[i, 1], Subscript[i, 1]] Pochhammer[1/2, Subscript[i, 2]] Pochhammer[1/2, Subscript[i, 3]] (1 - n Sin[Subscript[z, 0]]^2)^(-1 - Subscript[i, 1]) Cos[Subscript[z, 0]]^(-2 Subscript[i, 2] - 1) (1 - m Sin[Subscript[z, 0]]^2)^(-(1/2) - Subscript[i, 3]), { Subscript[i, 1], 0, i}, {Subscript[i, 2], 0, i}, { Subscript[i, 3], 0, i}], {i, 0, j - 1}])/ E^((1/2) I ((p - 2 Subscript[k, 2] - Subscript[k, 1] + j) Pi + 2 (2 Subscript[k, 2] + Subscript[k, 1] - j) Subscript[z, 0])), {Subscript[k, 2], 0, j - Subscript[k, 1]}], {Subscript[k, 1], 0, j - 1}], {j, 1, p}] (z - Subscript[z, 0])^p, {p, 1, Infinity}] /; Element[Re[Subscript[z, 0]/(2 Pi) - 3/4], Integers]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["EllipticPi", "[", RowBox[List["n", ",", "z", ",", "m"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", RowBox[List["(", RowBox[List[RowBox[List["2", RowBox[List["Re", "[", RowBox[List[FractionBox[SubscriptBox["z", "0"], RowBox[List["2", "\[Pi]"]]], "-", FractionBox["3", "4"]]], "]"]]]], "+", "1"]], ")"]], " ", RowBox[List["EllipticPi", "[", RowBox[List["n", ",", "m"]], "]"]]]], "+", RowBox[List[FractionBox["1", SqrtBox["m"]], RowBox[List["EllipticPi", "[", RowBox[List[FractionBox["n", "m"], ",", FractionBox["1", "m"]]], "]"]]]]]], ")"]], RowBox[List["(", RowBox[List["1", "-", RowBox[List["Exp", "[", RowBox[List[RowBox[List["-", "\[Pi]"]], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "4"], "+", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["z", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]], "+", RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "4"], "-", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["z", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]]]], ")"]]]], "]"]]]], ")"]]]], "+", RowBox[List[RowBox[List["Exp", "[", RowBox[List[RowBox[List["-", "\[Pi]"]], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "4"], "+", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["z", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]], "+", RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "4"], "-", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["z", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]]]], ")"]]]], "]"]], RowBox[List["EllipticPi", "[", RowBox[List["n", ",", SubscriptBox["z", "0"], ",", "m"]], "]"]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["p", "=", "1"]], "\[Infinity]"], RowBox[List[FractionBox["1", RowBox[List["p", "!"]]], RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "p"], RowBox[List[FractionBox["1", RowBox[List["j", "!"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", "1"], "=", "0"]], RowBox[List["j", "-", "1"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["j", ",", SubscriptBox["k", "1"]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", "2"], "=", "0"]], RowBox[List["j", "-", SubscriptBox["k", "1"]]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], SubscriptBox["k", "1"]], SuperscriptBox["2", RowBox[List[SubscriptBox["k", "1"], "-", "j"]]], " ", SuperscriptBox[RowBox[List["Sin", "[", SubscriptBox["z", "0"], "]"]], SubscriptBox["k", "1"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", SubscriptBox["k", "2"]]], "+", SubscriptBox["k", "1"], "-", "j"]], ")"]], "p"], SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["p", "-", RowBox[List["2", " ", SubscriptBox["k", "2"]]], "-", SubscriptBox["k", "1"], "+", "j"]], ")"]], " ", "\[Pi]"]], "+", RowBox[List["2", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SubscriptBox["k", "2"]]], "+", SubscriptBox["k", "1"], "-", "j"]], ")"]], SubscriptBox["z", "0"]]]]], ")"]]]]], RowBox[List["Binomial", "[", RowBox[List[RowBox[List["j", "-", SubscriptBox["k", "1"]]], ",", SubscriptBox["k", "2"]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], RowBox[List["j", "-", "1"]]], RowBox[List["(", RowBox[List[FractionBox[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "j"]], ",", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["j", "-", "i"]], ")"]]]], "-", "2"]]]], "]"]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["j", "-", "i", "-", "1"]], ")"]], "!"]], SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", RowBox[List["Sin", "[", SubscriptBox["z", "0"], "]"]]]], ")"]], RowBox[List["j", "-", RowBox[List["2", " ", "i"]], "-", "1"]]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["i", "1"], "=", "0"]], "i"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["i", "2"], "=", "0"]], "i"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["i", "3"], "=", "0"]], "i"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], SubscriptBox["i", "1"]], RowBox[List["KroneckerDelta", "[", RowBox[List[SubscriptBox["i", "1"], "+", SubscriptBox["i", "2"], "+", SubscriptBox["i", "3"], "-", "i"]], "]"]], " ", RowBox[List["Multinomial", "[", RowBox[List[SubscriptBox["i", "1"], ",", SubscriptBox["i", "2"], ",", SubscriptBox["i", "3"]]], "]"]], SuperscriptBox["n", SubscriptBox["i", "1"]], SuperscriptBox["m", SubscriptBox["i", "3"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", SubscriptBox["i", "1"]]], ",", SubscriptBox["i", "1"]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", SubscriptBox["i", "2"]]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", SubscriptBox["i", "3"]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["n", " ", SuperscriptBox[RowBox[List["Sin", "[", SubscriptBox["z", "0"], "]"]], "2"]]]]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", SubscriptBox["i", "1"]]]], " ", SuperscriptBox[RowBox[List["Cos", "[", SubscriptBox["z", "0"], "]"]], RowBox[List[RowBox[List[RowBox[List["-", "2"]], SubscriptBox["i", "2"]]], "-", "1"]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", SubscriptBox["z", "0"], "]"]], "2"]]]]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", SubscriptBox["i", "3"]]]]]]]]]]]]]], ")"]]]]]]]]]]]]]]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]], "p"]]]]]]]]], "/;", RowBox[List[RowBox[List["Re", "[", RowBox[List[FractionBox[SubscriptBox["z", "0"], RowBox[List["2", "\[Pi]"]]], "-", FractionBox["3", "4"]]], "]"]], "\[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> &#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> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <msqrt> <mi> m </mi> </msqrt> </mfrac> <mo> &#8290; </mo> <mrow> <mi> &#928; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mi> n </mi> <mi> m </mi> </mfrac> <mo> &#10072; </mo> <mfrac> <mn> 1 </mn> <mi> m </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mfrac> <mo> - </mo> <mfrac> <mn> 3 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#928; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> - </mo> <mi> &#960; </mi> </mrow> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> &#8970; </mo> <mrow> <mfrac> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mfrac> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> &#8971; </mo> </mrow> <mo> + </mo> <mrow> <mo> &#8970; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> - </mo> <mfrac> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mfrac> </mrow> <mo> &#8971; </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> - </mo> <mi> &#960; </mi> </mrow> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> &#8970; </mo> <mrow> <mfrac> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mfrac> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> &#8971; </mo> </mrow> <mo> + </mo> <mrow> <mo> &#8970; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> - </mo> <mfrac> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mfrac> </mrow> <mo> &#8971; </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> &#928; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> ; </mo> <mrow> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> &#10072; </mo> <mi> m </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> p </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mi> p </mi> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> p </mi> </munderover> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mi> j </mi> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> k </mi> <mn> 1 </mn> </msub> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> j </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mi> j </mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi> k </mi> <mn> 1 </mn> </msub> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, GridBox[List[List[TagBox[&quot;j&quot;, Identity, Rule[Editable, True]]], List[TagBox[SubscriptBox[&quot;k&quot;, &quot;1&quot;], Identity, Rule[Editable, True]]]]], &quot;)&quot;]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> k </mi> <mn> 2 </mn> </msub> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> j </mi> <mo> - </mo> <msub> <mi> k </mi> <mn> 1 </mn> </msub> </mrow> </munderover> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <msub> <mi> k </mi> <mn> 1 </mn> </msub> </msup> <mo> &#8290; </mo> <msup> <mn> 2 </mn> <mrow> <msub> <mi> k </mi> <mn> 1 </mn> </msub> <mo> - </mo> <mi> j </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <msup> <mi> sin </mi> <msub> <mi> k </mi> <mn> 1 </mn> </msub> </msup> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msub> <mi> k </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msub> <mi> k </mi> <mn> 2 </mn> </msub> </mrow> <mo> - </mo> <mi> j </mi> </mrow> <mo> ) </mo> </mrow> <mi> p </mi> </msup> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> j </mi> <mo> + </mo> <mi> p </mi> <mo> - </mo> <msub> <mi> k </mi> <mn> 1 </mn> </msub> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msub> <mi> k </mi> <mn> 2 </mn> </msub> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> k </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msub> <mi> k </mi> <mn> 2 </mn> </msub> </mrow> <mo> - </mo> <mi> j </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> &#8290; </mo> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mrow> <mi> j </mi> <mo> - </mo> <msub> <mi> k </mi> <mn> 1 </mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi> k </mi> <mn> 2 </mn> </msub> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, GridBox[List[List[TagBox[RowBox[List[&quot;j&quot;, &quot;-&quot;, SubscriptBox[&quot;k&quot;, &quot;1&quot;]]], Identity, Rule[Editable, True]]], List[TagBox[SubscriptBox[&quot;k&quot;, &quot;2&quot;], Identity, Rule[Editable, True]]]]], &quot;)&quot;]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> i </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> j </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <semantics> <mfrac> <msub> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> j </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> j </mi> <mo> - </mo> <mi> i </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 2 </mn> </mrow> </msub> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> j </mi> <mo> - </mo> <mi> i </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> j </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> i </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> </mrow> </mfrac> <annotation encoding='Mathematica'> TagBox[FractionBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;1&quot;, &quot;-&quot;, &quot;j&quot;]], &quot;)&quot;]], RowBox[List[RowBox[List[&quot;2&quot;, &quot; &quot;, RowBox[List[&quot;(&quot;, RowBox[List[&quot;j&quot;, &quot;-&quot;, &quot;i&quot;]], &quot;)&quot;]]]], &quot;-&quot;, &quot;2&quot;]]], RowBox[List[RowBox[List[RowBox[List[&quot;(&quot;, RowBox[List[&quot;j&quot;, &quot;-&quot;, &quot;i&quot;, &quot;-&quot;, &quot;1&quot;]], &quot;)&quot;]], &quot;!&quot;]], &quot; &quot;, SuperscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;2&quot;, &quot; &quot;, RowBox[List[&quot;sin&quot;, &quot;(&quot;, SubscriptBox[&quot;z&quot;, &quot;0&quot;], &quot;)&quot;]]]], &quot;)&quot;]], RowBox[List[&quot;j&quot;, &quot;-&quot;, RowBox[List[&quot;2&quot;, &quot;i&quot;]], &quot;-&quot;, &quot;1&quot;]]]]]], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> i </mi> <mn> 1 </mn> </msub> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> i </mi> </munderover> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> i </mi> <mn> 2 </mn> </msub> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> i </mi> </munderover> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> i </mi> <mn> 3 </mn> </msub> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> i </mi> </munderover> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <msub> <mi> i </mi> <mn> 1 </mn> </msub> </msup> <mo> &#8290; </mo> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <msub> <mi> i </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> i </mi> <mn> 2 </mn> </msub> <mo> + </mo> <msub> <mi> i </mi> <mn> 3 </mn> </msub> <mo> - </mo> <mi> i </mi> </mrow> </msub> <mo> &#8290; </mo> <semantics> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <msub> <mi> i </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> i </mi> <mn> 2 </mn> </msub> <mo> + </mo> <msub> <mi> i </mi> <mn> 3 </mn> </msub> </mrow> <mo> ; </mo> <msub> <mi> i </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <msub> <mi> i </mi> <mn> 2 </mn> </msub> <mo> , </mo> <msub> <mi> i </mi> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, RowBox[List[RowBox[List[RowBox[List[SubscriptBox[&quot;i&quot;, &quot;1&quot;], &quot;+&quot;, SubscriptBox[&quot;i&quot;, &quot;2&quot;], &quot;+&quot;, SubscriptBox[&quot;i&quot;, &quot;3&quot;]]], &quot;;&quot;, SubscriptBox[&quot;i&quot;, &quot;1&quot;]]], &quot;,&quot;, SubscriptBox[&quot;i&quot;, &quot;2&quot;], &quot;,&quot;, SubscriptBox[&quot;i&quot;, &quot;3&quot;]]], &quot;)&quot;]], Multinomial, Rule[Editable, False]] </annotation> </semantics> <mo> &#8290; </mo> <msup> <mi> n </mi> <msub> <mi> i </mi> <mn> 1 </mn> </msub> </msup> <mo> &#8290; </mo> <msup> <mi> m </mi> <msub> <mi> i </mi> <mn> 3 </mn> </msub> </msup> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <msub> <mi> i </mi> <mn> 1 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <msub> <mi> i </mi> <mn> 1 </mn> </msub> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;-&quot;, SubscriptBox[&quot;i&quot;, &quot;1&quot;]]], &quot;)&quot;]], SubscriptBox[&quot;i&quot;, &quot;1&quot;]], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <msub> <mi> i </mi> <mn> 2 </mn> </msub> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, FractionBox[&quot;1&quot;, &quot;2&quot;], &quot;)&quot;]], SubscriptBox[&quot;i&quot;, &quot;2&quot;]], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <msub> <mi> i </mi> <mn> 3 </mn> </msub> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, FractionBox[&quot;1&quot;, &quot;2&quot;], &quot;)&quot;]], SubscriptBox[&quot;i&quot;, &quot;3&quot;]], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> n </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> sin </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <msub> <mi> i </mi> <mn> 1 </mn> </msub> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <msup> <mi> cos </mi> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <msub> <mi> i </mi> <mn> 2 </mn> </msub> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> m </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> sin </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <msub> <mi> i </mi> <mn> 3 </mn> </msub> </mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msup> <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> p </mi> </msup> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mfrac> <mo> - </mo> <mfrac> <mn> 3 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[List[], Integers]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> EllipticPi </ci> <ci> n </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> m </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> EllipticPi </ci> <apply> <times /> <ci> n </ci> <apply> <power /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <real /> <apply> <plus /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 3 <sep /> 4 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <ci> EllipticPi </ci> <ci> n </ci> <ci> m </ci> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <pi /> </apply> <imaginaryi /> <apply> <plus /> <apply> <floor /> <apply> <plus /> <apply> <times /> <apply> <arg /> <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> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 4 </cn> </apply> </apply> <apply> <floor /> <apply> <plus /> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <arg /> <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> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <pi /> </apply> <imaginaryi /> <apply> <plus /> <apply> <floor /> <apply> <plus /> <apply> <times /> <apply> <arg /> <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> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 4 </cn> </apply> </apply> <apply> <floor /> <apply> <plus /> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <arg /> <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> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <ci> EllipticPi </ci> <ci> n </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <ci> m </ci> </apply> </apply> <apply> <sum /> <bvar> <ci> p </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <factorial /> <ci> p </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> p </ci> </uplimit> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <factorial /> <ci> j </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </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> Binomial </ci> <ci> j </ci> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <sum /> <bvar> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 2 </cn> </apply> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> j </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </uplimit> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> </apply> <apply> <power /> <apply> <sin /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> <ci> p </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <imaginaryi /> <apply> <plus /> <apply> <times /> <pi /> <apply> <plus /> <ci> j </ci> <ci> p </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <ci> Binomial </ci> <apply> <plus /> <ci> j </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> i </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> Pochhammer </ci> <apply> <times /> <apply> <ci> Subscript </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> j </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> i </ci> </apply> </apply> </apply> <cn type='integer'> -2 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <apply> <plus /> <ci> j </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> i </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <sin /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <apply> <plus /> <ci> j </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> i </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <sum /> <bvar> <apply> <ci> Subscript </ci> <ci> i </ci> <cn type='integer'> 3 </cn> </apply> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> i </ci> </uplimit> <apply> <sum /> <bvar> <apply> <ci> Subscript </ci> <ci> i </ci> <cn type='integer'> 2 </cn> </apply> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> i </ci> </uplimit> <apply> <sum /> <bvar> <apply> <ci> Subscript </ci> <ci> i </ci> <cn type='integer'> 1 </cn> </apply> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> i </ci> </uplimit> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> i </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> KroneckerDelta </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> i </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> i </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> i </ci> <cn type='integer'> 3 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> i </ci> </apply> </apply> </apply> <apply> <ci> Multinomial </ci> <apply> <ci> Subscript </ci> <ci> i </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> i </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> i </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <power /> <ci> n </ci> <apply> <ci> Subscript </ci> <ci> i </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <ci> m </ci> <apply> <ci> Subscript </ci> <ci> i </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <ci> Pochhammer </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> i </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> i </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Pochhammer </ci> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <ci> Subscript </ci> <ci> i </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ci> Pochhammer </ci> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <ci> Subscript </ci> <ci> i </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> n </ci> <apply> <power /> <apply> <sin /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> i </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <cos /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <apply> <ci> Subscript </ci> <ci> i </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> m </ci> <apply> <power /> <apply> <sin /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> i </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </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> p </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <apply> <real /> <apply> <plus /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 3 <sep /> 4 </cn> </apply> </apply> </apply> <integers /> </apply> </apply> </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[RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["Re", "[", RowBox[List[FractionBox[SubscriptBox["zz", "0"], RowBox[List["2", " ", "\[Pi]"]]], "-", FractionBox["3", "4"]]], "]"]]]], "+", "1"]], ")"]], " ", RowBox[List["EllipticPi", "[", RowBox[List["n", ",", "m"]], "]"]]]], "+", FractionBox[RowBox[List["EllipticPi", "[", RowBox[List[FractionBox["n", "m"], ",", FractionBox["1", "m"]]], "]"]], SqrtBox["m"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[Pi]"]], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "4"], "+", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]], "+", RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "4"], "-", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]]]], ")"]]]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[Pi]"]], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "4"], "+", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]], "+", RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "4"], "-", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]]]], ")"]]]]], " ", RowBox[List["EllipticPi", "[", RowBox[List["n", ",", SubscriptBox["zz", "0"], ",", "m"]], "]"]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["p", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "p"], FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", "1"], "=", "0"]], RowBox[List["j", "-", "1"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["j", ",", SubscriptBox["k", "1"]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", "2"], "=", "0"]], RowBox[List["j", "-", SubscriptBox["k", "1"]]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], SubscriptBox["k", "1"]], " ", SuperscriptBox["2", RowBox[List[SubscriptBox["k", "1"], "-", "j"]]], " ", SuperscriptBox[RowBox[List["Sin", "[", SubscriptBox["zz", "0"], "]"]], SubscriptBox["k", "1"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", SubscriptBox["k", "2"]]], "+", SubscriptBox["k", "1"], "-", "j"]], ")"]], "p"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["p", "-", RowBox[List["2", " ", SubscriptBox["k", "2"]]], "-", SubscriptBox["k", "1"], "+", "j"]], ")"]], " ", "\[Pi]"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SubscriptBox["k", "2"]]], "+", SubscriptBox["k", "1"], "-", "j"]], ")"]], " ", SubscriptBox["zz", "0"]]]]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["j", "-", SubscriptBox["k", "1"]]], ",", SubscriptBox["k", "2"]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], RowBox[List["j", "-", "1"]]], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "j"]], ",", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["j", "-", "i"]], ")"]]]], "-", "2"]]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["i", "1"], "=", "0"]], "i"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["i", "2"], "=", "0"]], "i"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["i", "3"], "=", "0"]], "i"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], SubscriptBox["i", "1"]], " ", RowBox[List["KroneckerDelta", "[", RowBox[List[SubscriptBox["i", "1"], "+", SubscriptBox["i", "2"], "+", SubscriptBox["i", "3"], "-", "i"]], "]"]], " ", RowBox[List["Multinomial", "[", RowBox[List[SubscriptBox["i", "1"], ",", SubscriptBox["i", "2"], ",", SubscriptBox["i", "3"]]], "]"]], " ", SuperscriptBox["n", SubscriptBox["i", "1"]], " ", SuperscriptBox["m", SubscriptBox["i", "3"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", SubscriptBox["i", "1"]]], ",", SubscriptBox["i", "1"]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", SubscriptBox["i", "2"]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", SubscriptBox["i", "3"]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["n", " ", SuperscriptBox[RowBox[List["Sin", "[", SubscriptBox["zz", "0"], "]"]], "2"]]]]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", SubscriptBox["i", "1"]]]], " ", SuperscriptBox[RowBox[List["Cos", "[", SubscriptBox["zz", "0"], "]"]], RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", SubscriptBox["i", "2"]]], "-", "1"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", SubscriptBox["zz", "0"], "]"]], "2"]]]]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", SubscriptBox["i", "3"]]]]]]]]]]]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["j", "-", "i", "-", "1"]], ")"]], "!"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", RowBox[List["Sin", "[", SubscriptBox["zz", "0"], "]"]]]], ")"]], RowBox[List["j", "-", RowBox[List["2", " ", "i"]], "-", "1"]]]]]]]]]]]]]]]], RowBox[List["j", "!"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]], "p"]]], RowBox[List["p", "!"]]]]]]], "/;", RowBox[List[RowBox[List["Re", "[", RowBox[List[FractionBox[SubscriptBox["zz", "0"], RowBox[List["2", " ", "\[Pi]"]]], "-", FractionBox["3", "4"]]], "]"]], "\[Element]", "Integers"]]]]]]]]










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





2007-05-02