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ArcSinh






Mathematica Notation

Traditional Notation









Elementary Functions > ArcSinh[z] > Series representations > Generalized power series > Expansions at z==-i > For small integer powers of the function > For the second power





http://functions.wolfram.com/01.25.06.0055.01









  


  










Input Form





ArcSinh[z]^2 \[Proportional] -(Pi^2/4) + Pi Sqrt[2] Sqrt[(-I) (z + I)] (1 - (1/12) I (z + I) - (3/160) (z + I)^2 + \[Ellipsis]) + 2 I (z + I) (1 - (1/6) I (z + I) - (2/45) (z + I)^2 + \[Ellipsis]) /; (z -> -I)










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["ArcSinh", "[", "z", "]"]], "2"], "\[Proportional]", RowBox[List[RowBox[List["-", FractionBox[SuperscriptBox["\[Pi]", "2"], "4"]]], "+", RowBox[List["\[Pi]", SqrtBox["2"], " ", SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], RowBox[List["(", RowBox[List["z", "+", "\[ImaginaryI]"]], ")"]]]]], RowBox[List["(", RowBox[List["1", "-", RowBox[List[FractionBox["1", "12"], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["z", "+", "\[ImaginaryI]"]], ")"]]]], "-", RowBox[List[FractionBox["3", "160"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "\[ImaginaryI]"]], ")"]], "2"]]], "+", "\[Ellipsis]"]], ")"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", RowBox[List["(", RowBox[List["z", "+", "\[ImaginaryI]"]], ")"]], RowBox[List["(", RowBox[List["1", "-", RowBox[List[FractionBox["1", "6"], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["z", "+", "\[ImaginaryI]"]], ")"]]]], "-", RowBox[List[FractionBox["2", "45"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "\[ImaginaryI]"]], ")"]], "2"]]], "+", "\[Ellipsis]"]], ")"]]]]]]]], "/;", RowBox[List["(", RowBox[List["z", "\[Rule]", RowBox[List["-", "\[ImaginaryI]"]]]], ")"]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msup> <mrow> <msup> <mi> sinh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8733; </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> <mn> 4 </mn> </mfrac> </mrow> <mo> + </mo> <mrow> <msqrt> <mn> 2 </mn> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mi> &#8520; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </msqrt> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 12 </mn> </mfrac> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mi> &#8520; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mn> 3 </mn> <mn> 160 </mn> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mi> &#8520; </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mo> &#8230; </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mi> &#8520; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 6 </mn> </mfrac> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mi> &#8520; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mn> 2 </mn> <mn> 45 </mn> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mi> &#8520; </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mo> &#8230; </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <power /> <apply> <arcsinh /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <apply> <plus /> <ci> z </ci> <imaginaryi /> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <pi /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 12 </cn> <imaginaryi /> <apply> <plus /> <ci> z </ci> <imaginaryi /> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='rational'> 3 <sep /> 160 </cn> <apply> <power /> <apply> <plus /> <ci> z </ci> <imaginaryi /> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <ci> &#8230; </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <apply> <plus /> <ci> z </ci> <imaginaryi /> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 6 </cn> <imaginaryi /> <apply> <plus /> <ci> z </ci> <imaginaryi /> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='rational'> 2 <sep /> 45 </cn> <apply> <power /> <apply> <plus /> <ci> z </ci> <imaginaryi /> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <ci> &#8230; </ci> </apply> </apply> </apply> </apply> <apply> <ci> Rule </ci> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", SuperscriptBox[RowBox[List["ArcSinh", "[", "z_", "]"]], "2"], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[SuperscriptBox["\[Pi]", "2"], "4"]]], "+", RowBox[List["\[Pi]", " ", SqrtBox["2"], " ", SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List["z", "+", "\[ImaginaryI]"]], ")"]]]]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List[FractionBox["1", "12"], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["z", "+", "\[ImaginaryI]"]], ")"]]]], "-", RowBox[List[FractionBox["3", "160"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "\[ImaginaryI]"]], ")"]], "2"]]], "+", "\[Ellipsis]"]], ")"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["z", "+", "\[ImaginaryI]"]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List[FractionBox["1", "6"], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["z", "+", "\[ImaginaryI]"]], ")"]]]], "-", RowBox[List[FractionBox["2", "45"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "\[ImaginaryI]"]], ")"]], "2"]]], "+", "\[Ellipsis]"]], ")"]]]]]], "/;", RowBox[List["(", RowBox[List["z", "\[Rule]", RowBox[List["-", "\[ImaginaryI]"]]]], ")"]]]]]]]]










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





2007-05-02





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