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ArcSech




Mathematica Notation

Traditional Notation









Elementary Functions > ArcSech[z] > Representations through equivalent functions > With related functions > Involving sinh-1 > Involving sech-1(a (b zc)m) > Involving sech-1(a (b zc)m) and sinh-1(i/ab-m z-m c)





http://functions.wolfram.com/01.30.27.1636.01









  


  








Input Form





ArcSech[a (b z^c)^m] == (Sqrt[1/a/(b z^c)^m - 1]/Sqrt[1 - 1/a/(b z^c)^m]) (Pi/2 + ((I b^m z^(m c))/(b z^c)^m) ArcSinh[I/a/(b^m z^(m c))]) /; Element[2 m, Integers]








Standard Form





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MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msup> <mi> sech </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> c </mi> </msup> </mrow> <mo> ) </mo> </mrow> <mi> m </mi> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> c </mi> </msup> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> m </mi> </mrow> </msup> <mi> a </mi> </mfrac> </mrow> </msqrt> </mfrac> <mo> &#8290; </mo> <msqrt> <mrow> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> c </mi> </msup> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> m </mi> </mrow> </msup> <mi> a </mi> </mfrac> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> <mo> + </mo> <mrow> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msup> <mi> b </mi> <mi> m </mi> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mi> m </mi> <mo> &#8290; </mo> <mi> c </mi> </mrow> </msup> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> c </mi> </msup> </mrow> <mo> ) </mo> </mrow> <mi> m </mi> </msup> </mfrac> <mo> &#8290; </mo> <mrow> <msup> <mi> sinh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mfrac> <mi> &#8520; </mi> <mi> a </mi> </mfrac> <mo> &#8290; </mo> <msup> <mi> b </mi> <mrow> <mo> - </mo> <mi> m </mi> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mrow> <mo> - </mo> <mi> m </mi> </mrow> <mo> &#8290; </mo> <mi> c </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> m </mi> </mrow> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <arcsech /> <apply> <times /> <ci> a </ci> <apply> <power /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> z </ci> <ci> c </ci> </apply> </apply> <ci> m </ci> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> z </ci> <ci> c </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> z </ci> <ci> c </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> b </ci> <ci> m </ci> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <ci> m </ci> <ci> c </ci> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> z </ci> <ci> c </ci> </apply> </apply> <ci> m </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <arcsinh /> <apply> <times /> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <ci> c </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> m </ci> </apply> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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Date Added to functions.wolfram.com (modification date)





2003-08-21