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
 











variants of this functions
KelvinKer






Mathematica Notation

Traditional Notation









Bessel-Type Functions > KelvinKer[nu,z] > Differentiation > Fractional integro-differentiation > With respect to z





http://functions.wolfram.com/03.20.20.0019.01









  


  










Input Form





D[KelvinKer[\[Nu], z], {z, \[Alpha]}] == Pi 2^(-Abs[\[Nu]] - 2) z^(Abs[\[Nu]] - \[Alpha]) Sum[((Sin[(Pi (2 (k + \[Nu]) + Abs[\[Nu]]))/4] Gamma[2 k + Abs[\[Nu]] + 1])/(k! (k + Abs[\[Nu]])! Gamma[2 k + Abs[\[Nu]] - \[Alpha] + 1])) (z/2)^(2 k), {k, 0, Infinity}] + (-1)^(Abs[\[Nu]] - 1) 2^(Abs[\[Nu]] - 2) z^(-Abs[\[Nu]] - \[Alpha]) Sum[(((E^((I Pi (2 \[Nu] + Abs[\[Nu]]))/4) + (-1)^k/E^((I Pi (2 \[Nu] + Abs[\[Nu]]))/4)) (Abs[\[Nu]] - k - 1)!)/ (k! (-2 k + Abs[\[Nu]] - 1)! Gamma[2 k - Abs[\[Nu]] - \[Alpha] + 1])) (Log[z] - PolyGamma[2 k - Abs[\[Nu]] - \[Alpha] + 1] + PolyGamma[-2 k + Abs[\[Nu]]]) ((I z^2)/4)^k, {k, 0, Floor[(Abs[\[Nu]] - 1)/2]}] + 2^(Abs[\[Nu]] - 2) z^(-Abs[\[Nu]] - \[Alpha]) Sum[(((E^((I Pi (2 \[Nu] + Abs[\[Nu]]))/4) + (-1)^k/E^((I Pi (2 \[Nu] + Abs[\[Nu]]))/4)) (Abs[\[Nu]] - k - 1)! Gamma[2 k - Abs[\[Nu]] + 1])/(k! Gamma[2 k - Abs[\[Nu]] - \[Alpha] + 1])) ((I z^2)/4)^k, {k, Floor[(Abs[\[Nu]] - 1)/2] + 1, Abs[\[Nu]] - 1}] - I^(\[Nu] + Abs[\[Nu]]) 2^(-1 - Abs[\[Nu]]) z^(Abs[\[Nu]] - \[Alpha]) Sum[((E^(-((I Pi Abs[\[Nu]])/4)) + (-1)^k E^((I Pi Abs[\[Nu]])/4))/ (k! (k + Abs[\[Nu]])!)) FDLogConstant[z, 2 k + Abs[\[Nu]], \[Alpha]] ((I z^2)/4)^k, {k, 0, Infinity}] + I^(\[Nu] + Abs[\[Nu]]) 2^(-2 - Abs[\[Nu]]) z^(Abs[\[Nu]] - \[Alpha]) Sum[(((E^(-((I Pi Abs[\[Nu]])/4)) + (-1)^k E^((I Pi Abs[\[Nu]])/4)) Gamma[2 k + Abs[\[Nu]] + 1])/(k! (k + Abs[\[Nu]])! Gamma[2 k + Abs[\[Nu]] - \[Alpha] + 1])) (2 Log[2] + PolyGamma[1 + k] + PolyGamma[1 + k + Abs[\[Nu]]]) ((I z^2)/4)^k, {k, 0, Infinity}] /; Element[\[Nu], Integers]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "\[Alpha]"]], "}"]]], RowBox[List["KelvinKer", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List["\[Pi]", " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], "-", "2"]]], SuperscriptBox["z", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "\[Alpha]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[" ", RowBox[List[RowBox[List["Sin", "[", FractionBox[RowBox[List["\[Pi]", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["k", "+", "\[Nu]"]], ")"]]]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]]]], "4"], "]"]], RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", "k"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]], "+", "1"]], "]"]]]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]], "!"]], RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", "k"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "\[Alpha]", "+", "1"]], "]"]]]]], SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List["2", " ", "k"]]]]]]]]], "+", " ", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "1"]]], SuperscriptBox["2", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "2"]]], SuperscriptBox["z", RowBox[List[RowBox[List["-", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], "-", "\[Alpha]"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", " ", FractionBox[RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "1"]], "2"], "]"]]], RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]]]], "4"]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]]]], "4"], " "]]]]]]], ")"]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "k", "-", "1"]], ")"]], "!"]]]], RowBox[List[RowBox[List["k", "!"]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "1"]], ")"]], "!"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", "k"]], "-", RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "\[Alpha]", "+", "1"]], "]"]]]]], RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "z", "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["2", " ", "k"]], "-", RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "\[Alpha]", "+", "1"]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], "]"]]]], ")"]], SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]], "4"], ")"]], "k"]]]]]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "2"]]], SuperscriptBox["z", RowBox[List[RowBox[List["-", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], "-", "\[Alpha]"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", RowBox[List[RowBox[List["Floor", "[", " ", FractionBox[RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "1"]], "2"], "]"]], "+", "1"]]]], RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "1"]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]]]], "4"]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]]]], "4"], " "]]]]]]], ")"]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "k", "-", "1"]], ")"]], "!"]], RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", "k"]], "-", RowBox[List["Abs", "[", "\[Nu]", "]"]], "+", "1"]], "]"]]]], " ", ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["k", "!"]], RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", "k"]], "-", RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "\[Alpha]", "+", "1"]], "]"]]]], ")"]]]], SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]], "4"], ")"]], "k"]]]]]]], "-", RowBox[List[SuperscriptBox["\[ImaginaryI]", RowBox[List["\[Nu]", "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "\[Alpha]"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], "4"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], "4"]]]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]], "!"]]]]], " ", RowBox[List["FDLogConstant", "[", RowBox[List["z", ",", RowBox[List[RowBox[List["2", " ", "k"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ",", "\[Alpha]"]], "]"]], SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]], "4"], ")"]], "k"]]]]]]], "+", RowBox[List[SuperscriptBox["\[ImaginaryI]", RowBox[List["\[Nu]", "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "\[Alpha]"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], "4"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], "4"]]]]]], ")"]], RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", "k"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]], "+", "1"]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]], "!"]], RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", "k"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "\[Alpha]", "+", "1"]], "]"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", RowBox[List["Log", "[", "2", "]"]]]], "+", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "k"]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "k", "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], "]"]]]], ")"]], SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]], "4"], ")"]], "k"]]]]]]]]]]], "/;", RowBox[List["\[Nu]", "\[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> <mfrac> <mrow> <msup> <mo> &#8706; </mo> <mi> &#945; </mi> </msup> <mrow> <msub> <mi> ker </mi> <mi> &#957; </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> &#8706; </mo> <msup> <mi> z </mi> <mi> &#945; </mi> </msup> </mrow> </mfrac> <mo> &#63449; </mo> <mrow> <mrow> <msup> <mn> 2 </mn> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> &#957; </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> - </mo> <mn> 2 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mrow> <mo> - </mo> <mi> &#945; </mi> </mrow> <mo> - </mo> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> &#957; </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> </mrow> </msup> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mrow> <mrow> <mo> &#8970; </mo> <mfrac> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> &#957; </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> &#8971; </mo> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> &#957; </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mo> + </mo> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> &#957; </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> + </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mo> + </mo> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> &#957; </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> &#957; </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> - </mo> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> - </mo> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> &#957; </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> - </mo> <mi> &#945; </mi> <mo> - </mo> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> &#957; </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mn> 4 </mn> </mfrac> <mo> ) </mo> </mrow> <mi> k </mi> </msup> </mrow> </mrow> </mrow> <mo> + </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> &#957; </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mn> 2 </mn> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> &#957; </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> - </mo> <mn> 2 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mrow> <mo> - </mo> <mi> &#945; </mi> </mrow> <mo> - </mo> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> &#957; </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> </mrow> </msup> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mo> &#8970; </mo> <mfrac> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> &#957; </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> &#8971; </mo> </mrow> </munderover> <mrow> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mo> + </mo> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> &#957; </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> + </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mo> + </mo> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> &#957; </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> &#957; </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> - </mo> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> - </mo> <mi> &#945; </mi> <mo> - </mo> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> &#957; </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> &#957; </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> <mrow> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> &#957; </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> - </mo> <mi> &#945; </mi> <mo> - </mo> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> &#957; </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mn> 4 </mn> </mfrac> <mo> ) </mo> </mrow> <mi> k </mi> </msup> </mrow> </mrow> </mrow> <mo> + </mo> <mrow> <msup> <mn> 2 </mn> <mrow> <mrow> <mo> - </mo> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> &#957; </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> </mrow> <mo> - </mo> <mn> 2 </mn> </mrow> </msup> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> &#957; </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> - </mo> <mi> &#945; </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mfrac> <mrow> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> &#957; </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> &#957; </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> &#957; </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> - </mo> <mi> &#945; </mi> <mo> + </mo> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> &#957; </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> </msup> </mrow> </mrow> </mrow> <mo> - </mo> <mrow> <msup> <mi> &#8520; </mi> <mrow> <mi> &#957; </mi> <mo> + </mo> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> &#957; </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> </mrow> </msup> <mo> &#8290; </mo> <msup> <mn> 2 </mn> <mrow> <mrow> <mo> - </mo> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> &#957; </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> &#957; </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> - </mo> <mi> &#945; </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> &#957; </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> + </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> &#957; </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> </mrow> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msubsup> <mi> &#8497;&#119966; </mi> <mi> log </mi> <mrow> <mo> ( </mo> <mi> &#945; </mi> <mo> ) </mo> </mrow> </msubsup> <mo> ( </mo> <mrow> <mi> z </mi> <mo> , </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> &#957; </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> <mrow> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> &#957; </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mn> 4 </mn> </mfrac> <mo> ) </mo> </mrow> <mi> k </mi> </msup> </mrow> </mrow> </mrow> <mo> + </mo> <mrow> <msup> <mn> 2 </mn> <mrow> <mrow> <mo> - </mo> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> &#957; </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> </mrow> <mo> - </mo> <mn> 2 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> &#8520; </mi> <mrow> <mi> &#957; </mi> <mo> + </mo> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> &#957; </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> &#957; </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> - </mo> <mi> &#945; </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> &#957; </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> + </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> &#957; </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> </mrow> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> &#957; </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> &#957; </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> <mrow> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> &#957; </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> - </mo> <mi> &#945; </mi> <mo> + </mo> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> &#957; </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mn> 4 </mn> </mfrac> <mo> ) </mo> </mrow> <mi> k </mi> </msup> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> &#957; </mi> <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> <eq /> <apply> <partialdiff /> <bvar> <ci> z </ci> <degree> <ci> &#945; </ci> </degree> </bvar> <apply> <ci> KelvinKer </ci> <ci> &#957; </ci> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <abs /> <ci> &#957; </ci> </apply> <cn type='integer'> -2 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <abs /> <ci> &#957; </ci> </apply> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <apply> <plus /> <apply> <floor /> <apply> <times /> <apply> <plus /> <apply> <abs /> <ci> &#957; </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </lowlimit> <uplimit> <apply> <plus /> <apply> <abs /> <ci> &#957; </ci> </apply> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <times /> <apply> <times /> <apply> <plus /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='rational'> 1 <sep /> 4 </cn> <imaginaryi /> <pi /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#957; </ci> </apply> <apply> <abs /> <ci> &#957; </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <times /> <imaginaryi /> <pi /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#957; </ci> </apply> <apply> <abs /> <ci> &#957; </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <factorial /> <apply> <plus /> <apply> <abs /> <ci> &#957; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <abs /> <ci> &#957; </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <ci> k </ci> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <abs /> <ci> &#957; </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <ci> k </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <apply> <abs /> <ci> &#957; </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <abs /> <ci> &#957; </ci> </apply> <cn type='integer'> -2 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <abs /> <ci> &#957; </ci> </apply> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <floor /> <apply> <times /> <apply> <plus /> <apply> <abs /> <ci> &#957; </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </uplimit> <apply> <times /> <apply> <times /> <apply> <plus /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='rational'> 1 <sep /> 4 </cn> <imaginaryi /> <pi /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#957; </ci> </apply> <apply> <abs /> <ci> &#957; </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <times /> <imaginaryi /> <pi /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#957; </ci> </apply> <apply> <abs /> <ci> &#957; </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <factorial /> <apply> <plus /> <apply> <abs /> <ci> &#957; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <ln /> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <abs /> <ci> &#957; </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <ci> PolyGamma </ci> <apply> <plus /> <apply> <abs /> <ci> &#957; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <ci> k </ci> </apply> <apply> <factorial /> <apply> <plus /> <apply> <abs /> <ci> &#957; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <abs /> <ci> &#957; </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <ci> k </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <abs /> <ci> &#957; </ci> </apply> </apply> <cn type='integer'> -2 </cn> </apply> </apply> <pi /> <apply> <power /> <ci> z </ci> <apply> <plus /> <apply> <abs /> <ci> &#957; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <sin /> <apply> <times /> <cn type='rational'> 1 <sep /> 4 </cn> <pi /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> k </ci> <ci> &#957; </ci> </apply> </apply> <apply> <abs /> <ci> &#957; </ci> </apply> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <apply> <abs /> <ci> &#957; </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <ci> k </ci> </apply> <apply> <factorial /> <apply> <plus /> <ci> k </ci> <apply> <abs /> <ci> &#957; </ci> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> <apply> <abs /> <ci> &#957; </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <imaginaryi /> <apply> <plus /> <ci> &#957; </ci> <apply> <abs /> <ci> &#957; </ci> </apply> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <abs /> <ci> &#957; </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <apply> <abs /> <ci> &#957; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <plus /> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <times /> <imaginaryi /> <pi /> <apply> <abs /> <ci> &#957; </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='rational'> 1 <sep /> 4 </cn> <imaginaryi /> <pi /> <apply> <abs /> <ci> &#957; </ci> </apply> </apply> </apply> </apply> </apply> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> &#8497;&#119966; </ci> <ci> log </ci> </apply> <ci> &#945; </ci> </apply> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <apply> <abs /> <ci> &#957; </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <ci> k </ci> </apply> <apply> <factorial /> <apply> <plus /> <ci> k </ci> <apply> <abs /> <ci> &#957; </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <abs /> <ci> &#957; </ci> </apply> </apply> <cn type='integer'> -2 </cn> </apply> </apply> <apply> <power /> <imaginaryi /> <apply> <plus /> <ci> &#957; </ci> <apply> <abs /> <ci> &#957; </ci> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <apply> <abs /> <ci> &#957; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <plus /> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <times /> <imaginaryi /> <pi /> <apply> <abs /> <ci> &#957; </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='rational'> 1 <sep /> 4 </cn> <imaginaryi /> <pi /> <apply> <abs /> <ci> &#957; </ci> </apply> </apply> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <apply> <abs /> <ci> &#957; </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> k </ci> <apply> <abs /> <ci> &#957; </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <ci> k </ci> </apply> <apply> <factorial /> <apply> <plus /> <ci> k </ci> <apply> <abs /> <ci> &#957; </ci> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> <apply> <abs /> <ci> &#957; </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> &#957; </ci> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "\[Alpha]_"]], "}"]]]]], RowBox[List["KelvinKer", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["\[Pi]", " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], "-", "2"]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "\[Alpha]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Sin", "[", RowBox[List[FractionBox["1", "4"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["k", "+", "\[Nu]"]], ")"]]]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", "k"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]], "+", "1"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List["2", " ", "k"]]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]], "!"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", "k"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "\[Alpha]", "+", "1"]], "]"]]]]]]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "1"]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "2"]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], "-", "\[Alpha]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "1"]], ")"]]]], "]"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "4"], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]]]], ")"]]]]]]]]], ")"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "k", "-", "1"]], ")"]], "!"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "z", "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["2", " ", "k"]], "-", RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "\[Alpha]", "+", "1"]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]], "4"], ")"]], "k"]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "1"]], ")"]], "!"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", "k"]], "-", RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "\[Alpha]", "+", "1"]], "]"]]]]]]]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "2"]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], "-", "\[Alpha]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", RowBox[List[RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "1"]], ")"]]]], "]"]], "+", "1"]]]], RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "1"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "4"], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]]]], ")"]]]]]]]]], ")"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "k", "-", "1"]], ")"]], "!"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", "k"]], "-", RowBox[List["Abs", "[", "\[Nu]", "]"]], "+", "1"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]], "4"], ")"]], "k"]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", "k"]], "-", RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "\[Alpha]", "+", "1"]], "]"]]]]]]]]], "-", RowBox[List[SuperscriptBox["\[ImaginaryI]", RowBox[List["\[Nu]", "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "\[Alpha]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "4"], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]]]]]], ")"]], " ", RowBox[List["FDLogConstant", "[", RowBox[List["z", ",", RowBox[List[RowBox[List["2", " ", "k"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ",", "\[Alpha]"]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]], "4"], ")"]], "k"]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]], "!"]]]]]]]]], "+", RowBox[List[SuperscriptBox["\[ImaginaryI]", RowBox[List["\[Nu]", "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "\[Alpha]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "4"], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", "k"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]], "+", "1"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["Log", "[", "2", "]"]]]], "+", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "k"]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "k", "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]], "4"], ")"]], "k"]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]], "!"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", "k"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "\[Alpha]", "+", "1"]], "]"]]]]]]]]]]], "/;", RowBox[List["\[Nu]", "\[Element]", "Integers"]]]]]]]]










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





2007-05-02





© 1998- Wolfram Research, Inc.