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http://functions.wolfram.com/03.15.16.0009.01
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KelvinKei[Subscript[z, 1] + Subscript[z, 2]] ==
Sum[KelvinBei[k, Subscript[z, 2]] KelvinKer[-k, Subscript[z, 1]] +
KelvinBer[k, Subscript[z, 2]] KelvinKei[-k, Subscript[z, 1]],
{k, -Infinity, Infinity}] /; Abs[Subscript[z, 2]/Subscript[z, 1]] < 1
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["KelvinKei", "[", RowBox[List[SubscriptBox["z", "1"], "+", SubscriptBox["z", "2"]]], "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", RowBox[List["-", "\[Infinity]"]]]], "\[Infinity]"], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["KelvinBei", "[", RowBox[List["k", ",", SubscriptBox["z", "2"]]], "]"]], " ", RowBox[List["KelvinKer", "[", RowBox[List[RowBox[List["-", "k"]], ",", SubscriptBox["z", "1"]]], "]"]]]], "+", RowBox[List[RowBox[List["KelvinBer", "[", RowBox[List["k", ",", SubscriptBox["z", "2"]]], "]"]], " ", RowBox[List["KelvinKei", "[", RowBox[List[RowBox[List["-", "k"]], ",", SubscriptBox["z", "1"]]], "]"]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", FractionBox[SubscriptBox["z", "2"], SubscriptBox["z", "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> <mi> kei </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo>  </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mrow> <mo> - </mo> <mi> ∞ </mi> </mrow> </mrow> <mi> ∞ </mi> </munderover> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <msub> <mi> ber </mi> <mi> k </mi> </msub> <mo> ( </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <mi> kei </mi> <mrow> <mo> - </mo> <mi> k </mi> </mrow> </msub> <mo> ( </mo> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <msub> <mi> bei </mi> <mi> k </mi> </msub> <mo> ( </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <mi> ker </mi> <mrow> <mo> - </mo> <mi> k </mi> </mrow> </msub> <mo> ( </mo> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mfrac> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <msub> <mi> z </mi> <mn> 1 </mn> </msub> </mfrac> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> < </mo> <mn> 1 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> KelvinKei </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <apply> <times /> <cn type='integer'> -1 </cn> <infinity /> </apply> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <plus /> <apply> <times /> <apply> <ci> KelvinBer </ci> <ci> k </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ci> KelvinKei </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <ci> KelvinBei </ci> <ci> k </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ci> KelvinKer </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <lt /> <apply> <abs /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["KelvinKei", "[", RowBox[List[SubscriptBox["z_", "1"], "+", SubscriptBox["z_", "2"]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", RowBox[List["-", "\[Infinity]"]]]], "\[Infinity]"], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["KelvinBei", "[", RowBox[List["k", ",", SubscriptBox["zz", "2"]]], "]"]], " ", RowBox[List["KelvinKer", "[", RowBox[List[RowBox[List["-", "k"]], ",", SubscriptBox["zz", "1"]]], "]"]]]], "+", RowBox[List[RowBox[List["KelvinBer", "[", RowBox[List["k", ",", SubscriptBox["zz", "2"]]], "]"]], " ", RowBox[List["KelvinKei", "[", RowBox[List[RowBox[List["-", "k"]], ",", SubscriptBox["zz", "1"]]], "]"]]]]]], ")"]]]], "/;", RowBox[List[RowBox[List["Abs", "[", FractionBox[SubscriptBox["zz", "2"], SubscriptBox["zz", "1"]], "]"]], "<", "1"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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