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http://functions.wolfram.com/03.19.16.0018.01
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KelvinKei[\[Nu], Subscript[z, 1] Subscript[z, 2]] ==
Subscript[z, 1]^\[Nu]
Sum[(((1 - Subscript[z, 1]^2)^k (Subscript[z, 2]/2)^k)/k!)
(Sin[(3 k Pi)/4] KelvinKer[\[Nu] + k, Subscript[z, 2]] +
Cos[(3 k Pi)/4] KelvinKei[\[Nu] + k, Subscript[z, 2]]),
{k, 0, Infinity}] /; Abs[Subscript[z, 1]^2 - 1] < 1
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["KelvinKei", "[", RowBox[List["\[Nu]", ",", RowBox[List[SubscriptBox["z", "1"], " ", SubscriptBox["z", "2"]]]]], "]"]], "\[Equal]", RowBox[List[SubsuperscriptBox["z", "1", "\[Nu]"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SubsuperscriptBox["z", "1", "2"]]], ")"]], "k"], " ", SuperscriptBox[RowBox[List["(", FractionBox[SubscriptBox["z", "2"], "2"], ")"]], "k"]]], RowBox[List["k", "!"]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Sin", "[", FractionBox[RowBox[List["3", " ", "k", " ", "\[Pi]"]], "4"], "]"]], " ", RowBox[List["KelvinKer", "[", RowBox[List[RowBox[List["\[Nu]", "+", "k"]], ",", SubscriptBox["z", "2"]]], "]"]]]], "+", RowBox[List[RowBox[List["Cos", "[", FractionBox[RowBox[List["3", " ", "k", " ", "\[Pi]"]], "4"], "]"]], " ", RowBox[List["KelvinKei", "[", RowBox[List[RowBox[List["\[Nu]", "+", "k"]], ",", SubscriptBox["z", "2"]]], "]"]]]]]], ")"]]]]]]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", RowBox[List[SubsuperscriptBox["z", "1", "2"], "-", "1"]], "]"]], "<", "1"]]]]]]
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<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> ν </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <msub> <mi> ker </mi> <mi> ν </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mtext> </mtext> <mrow> <mi> csc </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mrow> <msup> <mi> ⅇ </mi> <mfrac> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mn> 4 </mn> </mfrac> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 4 </mn> </mrow> </msup> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mi> ν </mi> </msup> </mrow> <msup> <mi> z </mi> <mi> ν </mi> </msup> </mfrac> <mo> ⁢ </mo> <mrow> <msub> <mi> I </mi> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> </msub> <mo> ( </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 4 </mn> </mrow> </msup> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <msup> <mi> ⅇ </mi> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mn> 4 </mn> </mfrac> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> ν </mi> </msup> <mtext> </mtext> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 4 </mn> </mrow> </msup> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mi> ν </mi> </msup> </mfrac> <mo> ⁢ </mo> <mrow> <msub> <mi> I </mi> <mi> ν </mi> </msub> <mo> ( </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 4 </mn> </mrow> </msup> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <plus /> <apply> <ci> KelvinKei </ci> <ci> ν </ci> <ci> z </ci> </apply> <apply> <times /> <imaginaryi /> <apply> <ci> KelvinKer </ci> <ci> ν </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <pi /> <imaginaryi /> <apply> <csc /> <apply> <times /> <pi /> <ci> ν </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 3 </cn> <imaginaryi /> <pi /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <cn type='rational'> 3 <sep /> 4 </cn> </apply> <ci> z </ci> </apply> <ci> ν </ci> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <ci> ν </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> BesselI </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <cn type='rational'> 3 <sep /> 4 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <pi /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <ci> ν </ci> </apply> <apply> <power /> <apply> <power /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <cn type='rational'> 3 <sep /> 4 </cn> </apply> <ci> z </ci> </apply> <ci> ν </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> BesselI </ci> <ci> ν </ci> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <cn type='rational'> 3 <sep /> 4 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["KelvinKei", "[", RowBox[List["\[Nu]_", ",", RowBox[List[SubscriptBox["z_", "1"], " ", SubscriptBox["z_", "2"]]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SubsuperscriptBox["zz", "1", "\[Nu]"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SubsuperscriptBox["zz", "1", "2"]]], ")"]], "k"], " ", SuperscriptBox[RowBox[List["(", FractionBox[SubscriptBox["zz", "2"], "2"], ")"]], "k"]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Sin", "[", FractionBox[RowBox[List["3", " ", "k", " ", "\[Pi]"]], "4"], "]"]], " ", RowBox[List["KelvinKer", "[", RowBox[List[RowBox[List["\[Nu]", "+", "k"]], ",", SubscriptBox["zz", "2"]]], "]"]]]], "+", RowBox[List[RowBox[List["Cos", "[", FractionBox[RowBox[List["3", " ", "k", " ", "\[Pi]"]], "4"], "]"]], " ", RowBox[List["KelvinKei", "[", RowBox[List[RowBox[List["\[Nu]", "+", "k"]], ",", SubscriptBox["zz", "2"]]], "]"]]]]]], ")"]]]], RowBox[List["k", "!"]]]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", RowBox[List[SubsuperscriptBox["zz", "1", "2"], "-", "1"]], "]"]], "<", "1"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
