Cosh[Sqrt[a] z] == (1/Sqrt[a]) MathieuSPrime[a, 0, I z]

Cell[BoxData[RowBox[List[RowBox[List["Cosh", "[", RowBox[List[SqrtBox["a"], " ", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", SqrtBox["a"]], RowBox[List["MathieuSPrime", "[", RowBox[List["a", ",", "0", ",", RowBox[List["\[ImaginaryI]", " ", "z"]]]], "]"]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mi> cosh </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> <mfrac> <mn> 1 </mn> <msqrt> <mi> a </mi> </msqrt> </mfrac> <mo> &#8290; </mo> <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> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <cosh /> <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> <apply> <power /> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <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> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Cosh", "[", RowBox[List[SqrtBox["a_"], " ", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["MathieuSPrime", "[", RowBox[List["a", ",", "0", ",", RowBox[List["\[ImaginaryI]", " ", "z"]]]], "]"]], SqrtBox["a"]]]]]]

2001-10-29