Coth[Sqrt[a] z] == (-I) (MathieuSPrime[a, 0, I z]/MathieuCPrime[a, 0, I z])

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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> <mi> coth </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msqrt> <mi> a </mi> </msqrt> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> &#8290; </mo> <mfrac> <mrow> <msub> <mi> Se </mi> <mi> z </mi> </msub> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mn> 0 </mn> <mo> , </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <msub> <mi> Ce </mi> <mi> z </mi> </msub> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mn> 0 </mn> <mo> , </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <coth /> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <apply> <times /> <apply> <apply> <ci> Subscript </ci> <ci> Se </ci> <ci> z </ci> </apply> <ci> a </ci> <cn type='integer'> 0 </cn> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <apply> <ci> Subscript </ci> <ci> Ce </ci> <ci> z </ci> </apply> <ci> a </ci> <cn type='integer'> 0 </cn> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2001-10-29