Kelvin function of the second kind: Representations through more general functions (formula 03.16.26.0042)
http://functions.wolfram.com/03.16.26.0042.01
Hypergeometric0F1[1, (I z^2)/4] KelvinKer[z] == (1/8) Sqrt[Pi] (MeijerG[{{}, {}}, {{0, 0}, {0, 1/2}}, z/(2 Sqrt[2]), 1/4] + I MeijerG[{{}, {}}, {{0, 1/2}, {0, 0}}, z/(2 Sqrt[2]), 1/4] + (1/(Sqrt[2] Pi)) (MeijerG[{{1/4, 3/4}, {}}, {{0, 0, 1/2}, {0, 1/2, 1/2}}, z/2, 1/4] - I MeijerG[{{1/4, 3/4}, {}}, {{0, 1/2, 1/2}, {0, 0, 1/2}}, z/2, 1/4]))
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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> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 0 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mo> &#8202; </mo> <mo> ; </mo> <mn> 1 </mn> <mo> ; </mo> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mn> 4 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, &quot;0&quot;], SubscriptBox[&quot;F&quot;, &quot;1&quot;]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[&quot;\[Null]&quot;, InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric0F1, Rule[Editable, False], Rule[Selectable, False]], &quot;;&quot;, 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<power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> KelvinKer </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 8 </cn> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <ci> MeijerG </ci> <list> <list /> <list /> </list> <list> <list> <cn type='integer'> 0 </cn> <cn type='integer'> 0 </cn> </list> <list> <cn type='integer'> 0 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </list> </list> <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> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <times /> <imaginaryi /> <apply> <ci> MeijerG </ci> <list> <list /> <list /> </list> <list> <list> <cn type='integer'> 0 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </list> <list> <cn type='integer'> 0 </cn> <cn type='integer'> 0 </cn> </list> </list> <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> <cn type='rational'> 1 <sep /> 4 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <ci> MeijerG </ci> <list> <list> <cn type='rational'> 1 <sep /> 4 </cn> <cn type='rational'> 3 <sep /> 4 </cn> </list> <list /> </list> <list> <list> <cn type='integer'> 0 </cn> <cn type='integer'> 0 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </list> <list> <cn type='integer'> 0 </cn> <cn type='rational'> 1 <sep /> 2 </cn> 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Kelvin function of the second kind: Representations through more general functions (formula 03.16.26.0042)
http://functions.wolfram.com/03.16.26.0042.01
Hypergeometric0F1[1, (I z^2)/4] KelvinKer[z] == (1/8) Sqrt[Pi] (MeijerG[{{}, {}}, {{0, 0}, {0, 1/2}}, z/(2 Sqrt[2]), 1/4] + I MeijerG[{{}, {}}, {{0, 1/2}, {0, 0}}, z/(2 Sqrt[2]), 1/4] + (1/(Sqrt[2] Pi)) (MeijerG[{{1/4, 3/4}, {}}, {{0, 0, 1/2}, {0, 1/2, 1/2}}, z/2, 1/4] - I MeijerG[{{1/4, 3/4}, {}}, {{0, 1/2, 1/2}, {0, 0, 1/2}}, z/2, 1/4]))
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