ArcSinh[z] == I InverseJacobiNS[I/z, 0]

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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> <msup> <mi> sinh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> ns </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mfrac> <mi> &#8520; </mi> <mi> z </mi> </mfrac> <mo> &#10072; </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <arcsinh /> <ci> z </ci> </apply> <apply> <times /> <imaginaryi /> <apply> <ci> InverseJacobiNS </ci> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcSinh", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List["\[ImaginaryI]", " ", RowBox[List["InverseJacobiNS", "[", RowBox[List[FractionBox["\[ImaginaryI]", "z"], ",", "0"]], "]"]]]]]]]]

2001-10-29