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

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











variants of this functions
KelvinKei




Mathematica Notation

Traditional Notation









Bessel-Type Functions > KelvinKei[nu,z] > Series representations > Generalized power series > Expansions at generic point z==z0





http://functions.wolfram.com/03.19.06.0008.01









  


  








Input Form





KelvinKei[\[Nu], z] == Sum[(1/(Subscript[z, 0]^k k!)) Sum[(-1)^(m + k) Binomial[k, m] Pochhammer[-\[Nu], k - m] Sum[(((-1)^i 2^(2 i - m) Pochhammer[-m, 2 (m - i)] Pochhammer[\[Nu], i])/(m - i)!) (KelvinKei[\[Nu], z] Sum[(((-1)^j (i - 2 j)!)/((2 j)! (i - 4 j)! Pochhammer[1 - i - \[Nu], 2 j] Pochhammer[\[Nu], 2 j])) (z/2)^(4 j), {j, 0, Floor[i/2]}] + (z/(2 Sqrt[2])) (KelvinKei[-1 + \[Nu], z] - KelvinKer[-1 + \[Nu], z]) Sum[(((-1)^j (i - 2 j - 1)!)/ ((2 j)! (i - 4 j - 1)! Pochhammer[1 - i - \[Nu], 2 j] Pochhammer[\[Nu], 2 j + 1])) (z/2)^(4 j), {j, 0, Floor[(i - 1)/2]}] - (z^2/4) KelvinKer[\[Nu], z] Sum[(((-1)^j (i - 2 j - 1)!)/((2 j + 1)! (i - 4 j - 2)! Pochhammer[1 - i - \[Nu], 2 j + 1] Pochhammer[\[Nu], 2 j + 1])) (z/2)^(4 j), {j, 0, Floor[(i - 1)/2]}] - (z^3/(8 Sqrt[2])) (KelvinKei[-1 + \[Nu], z] + KelvinKer[-1 + \[Nu], z]) Sum[(((-1)^j (i - 2 j - 2)!)/((2 j + 1)! (i - 4 j - 3)! Pochhammer[1 - i - \[Nu], 2 j + 1] Pochhammer[\[Nu], 2 j + 2])) (z/2)^(4 j), {j, 0, Floor[(i - 2)/2]}]) (z - Subscript[z, 0])^k, {i, 0, m}], {m, 0, k}], {k, 0, Infinity}] /; Abs[Arg[Subscript[z, 0]]] < Pi








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["KelvinKei", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[SubsuperscriptBox["z", "0", RowBox[List["-", "k"]]], RowBox[List["k", "!"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["m", "+", "k"]]], RowBox[List["Binomial", "[", RowBox[List["k", ",", "m"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["k", "-", "m"]]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], "m"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "i"], " ", SuperscriptBox["2", RowBox[List[RowBox[List["2", " ", "i"]], "-", "m"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "m"]], ",", RowBox[List["2", " ", RowBox[List["(", RowBox[List["m", "-", "i"]], ")"]]]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List["\[Nu]", ",", "i"]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["m", "-", "i"]], ")"]], "!"]]], RowBox[List["(", " ", RowBox[List[RowBox[List[RowBox[List["KelvinKei", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["Floor", "[", FractionBox["i", "2"], "]"]]], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], RowBox[List[RowBox[List["(", RowBox[List["i", "-", RowBox[List["2", "j"]]]], ")"]], "!"]], " "]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["2", "j"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["i", "-", RowBox[List["4", " ", "j"]]]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "i", "-", "\[Nu]"]], ",", RowBox[List["2", "j"]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List["\[Nu]", ",", RowBox[List["2", "j"]]]], "]"]]]]], SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List["4", "j"]]]]]]]]], "+", RowBox[List[FractionBox["z", RowBox[List["2", " ", SqrtBox["2"]]]], RowBox[List["(", RowBox[List[RowBox[List["KelvinKei", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], ",", "z"]], "]"]], "-", RowBox[List["KelvinKer", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], ",", "z"]], "]"]]]], ")"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["i", "-", "1"]], "2"], "]"]]], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], RowBox[List[RowBox[List["(", RowBox[List["i", "-", RowBox[List["2", "j"]], "-", "1"]], ")"]], "!"]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["2", "j"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["i", "-", RowBox[List["4", " ", "j"]], "-", "1"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "i", "-", "\[Nu]"]], ",", RowBox[List["2", "j"]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List["\[Nu]", ",", RowBox[List[RowBox[List["2", "j"]], "+", "1"]]]], "]"]]]]], " ", SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List["4", "j"]]]]]]]]], "-", RowBox[List[FractionBox[SuperscriptBox["z", "2"], "4"], RowBox[List["KelvinKer", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["i", "-", "1"]], "2"], "]"]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], RowBox[List[RowBox[List["(", RowBox[List["i", "-", RowBox[List["2", "j"]], "-", "1"]], ")"]], "!"]]]], " ", ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "j"]], "+", "1"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["i", "-", RowBox[List["4", " ", "j"]], "-", "2"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "i", "-", "\[Nu]"]], ",", RowBox[List[RowBox[List["2", "j"]], "+", "1"]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List["\[Nu]", ",", RowBox[List[RowBox[List["2", "j"]], "+", "1"]]]], "]"]]]], ")"]]]], SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List["4", "j"]]]]]]]]], "-", RowBox[List[FractionBox[SuperscriptBox["z", "3"], RowBox[List["8", " ", SqrtBox["2"]]]], RowBox[List["(", RowBox[List[RowBox[List["KelvinKei", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], ",", "z"]], "]"]], "+", RowBox[List["KelvinKer", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], ",", "z"]], "]"]]]], ")"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["i", "-", "2"]], "2"], "]"]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], RowBox[List[RowBox[List["(", RowBox[List["i", "-", RowBox[List["2", "j"]], "-", "2"]], ")"]], "!"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "j"]], "+", "1"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["i", "-", RowBox[List["4", " ", "j"]], "-", "3"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "i", "-", "\[Nu]"]], ",", RowBox[List[RowBox[List["2", "j"]], "+", "1"]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List["\[Nu]", ",", RowBox[List[RowBox[List["2", "j"]], "+", "2"]]]], "]"]]]], ")"]]]], " ", SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List["4", "j"]]]]]]]]]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]], "k"]]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", RowBox[List["Arg", "[", SubscriptBox["z", "0"], "]"]], "]"]], "<", "\[Pi]"]]]]]]








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> <msub> <mi> kei </mi> <mi> &#957; </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mfrac> <mrow> <msubsup> <mi> z </mi> <mn> 0 </mn> <mrow> <mo> - </mo> <mi> k </mi> </mrow> </msubsup> <mtext> </mtext> </mrow> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> m </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> k </mi> </munderover> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> k </mi> <mo> + </mo> <mi> m </mi> </mrow> </msup> <mo> &#8290; </mo> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mi> k </mi> </mtd> </mtr> <mtr> <mtd> <mi> m </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, GridBox[List[List[TagBox[&quot;k&quot;, Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox[&quot;m&quot;, Identity, Rule[Editable, True], Rule[Selectable, True]]]]], &quot;)&quot;]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> k </mi> <mo> - </mo> <mi> m </mi> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;-&quot;, &quot;\[Nu]&quot;]], &quot;)&quot;]], RowBox[List[&quot;k&quot;, &quot;-&quot;, &quot;m&quot;]]], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> i </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> m </mi> </munderover> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> i </mi> </msup> <mo> &#8290; </mo> <msup> <mn> 2 </mn> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> i </mi> </mrow> <mo> - </mo> <mi> m </mi> </mrow> </msup> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mi> i </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;-&quot;, &quot;m&quot;]], &quot;)&quot;]], RowBox[List[&quot;2&quot;, &quot; &quot;, RowBox[List[&quot;(&quot;, RowBox[List[&quot;m&quot;, &quot;-&quot;, &quot;i&quot;]], &quot;)&quot;]]]]], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mi> &#957; </mi> <mo> ) </mo> </mrow> <mi> i </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, &quot;\[Nu]&quot;, &quot;)&quot;]], &quot;i&quot;], Pochhammer] </annotation> </semantics> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mi> i </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mfrac> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mn> 4 </mn> </mfrac> </mrow> <mo> &#8290; </mo> <mtext> </mtext> <mrow> <msub> <mi> ker </mi> <mi> &#957; </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mo> &#8970; </mo> <mfrac> <mrow> <mi> i </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> &#8971; </mo> </mrow> </munderover> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> j </mi> </msup> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> i </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> j </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> j </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> i </mi> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> j </mi> </mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> i </mi> </mrow> <mo> - </mo> <mi> &#957; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> j </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[RowBox[List[&quot;-&quot;, &quot;i&quot;]], &quot;-&quot;, &quot;\[Nu]&quot;, &quot;+&quot;, &quot;1&quot;]], &quot;)&quot;]], RowBox[List[RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;j&quot;]], &quot;+&quot;, &quot;1&quot;]]], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mi> &#957; </mi> <mo> ) </mo> </mrow> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> j </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, &quot;\[Nu]&quot;, &quot;)&quot;]], RowBox[List[RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;j&quot;]], &quot;+&quot;, &quot;1&quot;]]], Pochhammer] </annotation> </semantics> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> j </mi> </mrow> </msup> </mrow> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <mi> z </mi> <mtext> </mtext> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mn> 2 </mn> </msqrt> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <mi> kei </mi> <mrow> <mi> &#957; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <msub> <mi> ker </mi> <mrow> <mi> &#957; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mo> &#8970; </mo> <mfrac> <mrow> <mi> i </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> &#8971; </mo> </mrow> </munderover> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> j </mi> </msup> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> i </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> j </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> j </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> i </mi> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> j </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> i </mi> </mrow> <mo> - </mo> <mi> &#957; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> j </mi> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[RowBox[List[&quot;-&quot;, &quot;i&quot;]], &quot;-&quot;, &quot;\[Nu]&quot;, &quot;+&quot;, &quot;1&quot;]], &quot;)&quot;]], RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;j&quot;]]], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mi> &#957; </mi> <mo> ) </mo> </mrow> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> j </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, &quot;\[Nu]&quot;, &quot;)&quot;]], RowBox[List[RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;j&quot;]], &quot;+&quot;, &quot;1&quot;]]], Pochhammer] </annotation> </semantics> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> j </mi> </mrow> </msup> </mrow> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <msup> <mi> z </mi> <mn> 3 </mn> </msup> <mtext> </mtext> </mrow> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <msqrt> <mn> 2 </mn> </msqrt> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <mi> kei </mi> <mrow> <mi> &#957; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <msub> <mi> ker </mi> <mrow> <mi> &#957; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mo> &#8970; </mo> <mfrac> <mrow> <mi> i </mi> <mo> - </mo> <mn> 2 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> &#8971; </mo> </mrow> </munderover> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> j </mi> </msup> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> i </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> j </mi> </mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> j </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> i </mi> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> j </mi> </mrow> <mo> - </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> i </mi> </mrow> <mo> - </mo> <mi> &#957; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> j </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[RowBox[List[&quot;-&quot;, &quot;i&quot;]], &quot;-&quot;, &quot;\[Nu]&quot;, &quot;+&quot;, &quot;1&quot;]], &quot;)&quot;]], RowBox[List[RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;j&quot;]], &quot;+&quot;, &quot;1&quot;]]], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mi> &#957; </mi> <mo> ) </mo> </mrow> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> j </mi> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, &quot;\[Nu]&quot;, &quot;)&quot;]], RowBox[List[RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;j&quot;]], &quot;+&quot;, &quot;2&quot;]]], Pochhammer] </annotation> </semantics> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> j </mi> </mrow> </msup> </mrow> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <msub> <mi> kei </mi> <mi> &#957; </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mo> &#8970; </mo> <mfrac> <mi> i </mi> <mn> 2 </mn> </mfrac> <mo> &#8971; </mo> </mrow> </munderover> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> j </mi> </msup> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> i </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> j </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> j </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> i </mi> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> j </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> i </mi> </mrow> <mo> - </mo> <mi> &#957; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> j </mi> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[RowBox[List[&quot;-&quot;, &quot;i&quot;]], &quot;-&quot;, &quot;\[Nu]&quot;, &quot;+&quot;, &quot;1&quot;]], &quot;)&quot;]], RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;j&quot;]]], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mi> &#957; </mi> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> j </mi> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, &quot;\[Nu]&quot;, &quot;)&quot;]], RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;j&quot;]]], Pochhammer] </annotation> </semantics> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> j </mi> </mrow> </msup> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &lt; </mo> <mi> &#960; </mi> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> KelvinKei </ci> <ci> &#957; </ci> <ci> z </ci> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> <apply> <power /> <apply> <factorial /> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> m </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> k </ci> </uplimit> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> k </ci> <ci> m </ci> </apply> </apply> <apply> <ci> Binomial </ci> <ci> k </ci> <ci> m </ci> </apply> <apply> <ci> Pochhammer </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> i </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> m </ci> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> i </ci> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> i </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> <apply> <ci> Pochhammer </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> i </ci> </apply> </apply> </apply> </apply> <apply> <ci> Pochhammer </ci> <ci> &#957; </ci> <ci> i </ci> </apply> <apply> <power /> <apply> <factorial /> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> i </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <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> </apply> <apply> <ci> KelvinKer </ci> <ci> &#957; </ci> <ci> z </ci> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <floor /> <apply> <times /> <apply> <plus /> <ci> i </ci> <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> <power /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <apply> <factorial /> <apply> <plus /> <ci> i </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> j </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> j </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <factorial /> <apply> <plus /> <ci> i </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <ci> j </ci> </apply> </apply> <cn type='integer'> -2 </cn> </apply> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> i </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> j </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Pochhammer </ci> <ci> &#957; </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> j </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'> 4 </cn> <ci> j </ci> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> kei </ci> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> -1 </cn> </apply> </apply> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> KelvinKer </ci> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> -1 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <floor /> <apply> <times /> <apply> <plus /> <ci> i </ci> <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> <power /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <apply> <factorial /> <apply> <plus /> <ci> i </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> j </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> j </ci> </apply> </apply> <apply> <factorial /> <apply> <plus /> <ci> i </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <ci> j </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> i </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> j </ci> </apply> </apply> <apply> <ci> Pochhammer </ci> <ci> &#957; </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> j </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'> 4 </cn> <ci> j </ci> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <ci> KelvinKei </ci> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> -1 </cn> </apply> <ci> z </ci> </apply> <apply> <ci> KelvinKer </ci> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> -1 </cn> </apply> <ci> z </ci> </apply> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <floor /> <apply> <times /> <apply> <plus /> <ci> i </ci> <cn type='integer'> -2 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <apply> <factorial /> <apply> <plus /> <ci> i </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> j </ci> </apply> </apply> <cn type='integer'> -2 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> j </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <factorial /> <apply> <plus /> <ci> i </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <ci> j </ci> </apply> </apply> <cn type='integer'> -3 </cn> </apply> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> i </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> j </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Pochhammer </ci> <ci> &#957; </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> j </ci> </apply> <cn type='integer'> 2 </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'> 4 </cn> <ci> j </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <ci> KelvinKei </ci> <ci> &#957; </ci> <ci> z </ci> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <floor /> <apply> <times /> <ci> i </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <apply> <factorial /> <apply> <plus /> <ci> i </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> j </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> j </ci> </apply> </apply> <apply> <factorial /> <apply> <plus /> <ci> i </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <ci> j </ci> </apply> </apply> </apply> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> i </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> j </ci> </apply> </apply> <apply> <ci> Pochhammer </ci> <ci> &#957; </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> j </ci> </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'> 4 </cn> <ci> j </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <lt /> <apply> <abs /> <apply> <arg /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <pi /> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["KelvinKei", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SubsuperscriptBox["zz", "0", RowBox[List["-", "k"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["m", "+", "k"]]], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "m"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["k", "-", "m"]]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], "m"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "i"], " ", SuperscriptBox["2", RowBox[List[RowBox[List["2", " ", "i"]], "-", "m"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "m"]], ",", RowBox[List["2", " ", RowBox[List["(", RowBox[List["m", "-", "i"]], ")"]]]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List["\[Nu]", ",", "i"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["KelvinKei", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["Floor", "[", FractionBox["i", "2"], "]"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List[RowBox[List["(", RowBox[List["i", "-", RowBox[List["2", " ", "j"]]]], ")"]], "!"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List["4", " ", "j"]]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["2", " ", "j"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["i", "-", RowBox[List["4", " ", "j"]]]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "i", "-", "\[Nu]"]], ",", RowBox[List["2", " ", "j"]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List["\[Nu]", ",", RowBox[List["2", " ", "j"]]]], "]"]]]]]]]]], "+", FractionBox[RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["KelvinKei", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], ",", "z"]], "]"]], "-", RowBox[List["KelvinKer", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], ",", "z"]], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["i", "-", "1"]], "2"], "]"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List[RowBox[List["(", RowBox[List["i", "-", RowBox[List["2", " ", "j"]], "-", "1"]], ")"]], "!"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List["4", " ", "j"]]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["2", " ", "j"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["i", "-", RowBox[List["4", " ", "j"]], "-", "1"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "i", "-", "\[Nu]"]], ",", RowBox[List["2", " ", "j"]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List["\[Nu]", ",", RowBox[List[RowBox[List["2", " ", "j"]], "+", "1"]]]], "]"]]]]]]]]], RowBox[List["2", " ", SqrtBox["2"]]]], "-", RowBox[List[FractionBox["1", "4"], " ", SuperscriptBox["z", "2"], " ", RowBox[List["KelvinKer", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["i", "-", "1"]], "2"], "]"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List[RowBox[List["(", RowBox[List["i", "-", RowBox[List["2", " ", "j"]], "-", "1"]], ")"]], "!"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List["4", " ", "j"]]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "j"]], "+", "1"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["i", "-", RowBox[List["4", " ", "j"]], "-", "2"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "i", "-", "\[Nu]"]], ",", RowBox[List[RowBox[List["2", " ", "j"]], "+", "1"]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List["\[Nu]", ",", RowBox[List[RowBox[List["2", " ", "j"]], "+", "1"]]]], "]"]]]]]]]]], "-", FractionBox[RowBox[List[SuperscriptBox["z", "3"], " ", RowBox[List["(", RowBox[List[RowBox[List["KelvinKei", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], ",", "z"]], "]"]], "+", RowBox[List["KelvinKer", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], ",", "z"]], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["i", "-", "2"]], "2"], "]"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List[RowBox[List["(", RowBox[List["i", "-", RowBox[List["2", " ", "j"]], "-", "2"]], ")"]], "!"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List["4", " ", "j"]]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "j"]], "+", "1"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["i", "-", RowBox[List["4", " ", "j"]], "-", "3"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "i", "-", "\[Nu]"]], ",", RowBox[List[RowBox[List["2", " ", "j"]], "+", "1"]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List["\[Nu]", ",", RowBox[List[RowBox[List["2", " ", "j"]], "+", "2"]]]], "]"]]]]]]]]], RowBox[List["8", " ", SqrtBox["2"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]], "k"]]], RowBox[List[RowBox[List["(", RowBox[List["m", "-", "i"]], ")"]], "!"]]]]]]]]]]], RowBox[List["k", "!"]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", RowBox[List["Arg", "[", SubscriptBox["zz", "0"], "]"]], "]"]], "<", "\[Pi]"]]]]]]]]


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





2007-05-02