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ArcSech




Mathematica Notation

Traditional Notation









Elementary Functions > ArcSech[z] > Series representations > Generalized power series > Expansions at z==1 > For small integer powers of the function > For the second power





http://functions.wolfram.com/01.30.06.0076.01









  


  








Input Form





ArcSech[z]^2 == -4 (z - 1) Sum[(((-1)^k Pochhammer[1/2, k])/k!) Hypergeometric2F1[1/2, 3/2 + k, 3/2, -1] (z - 1)^k, {k, 0, Infinity}]^ 2 /; Abs[z - 1] < 2








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["ArcSech", "[", "z", "]"]], "2"], "\[Equal]", RowBox[List[RowBox[List["-", "4"]], RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]]]], RowBox[List["k", "!"]]], RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox["1", "2"], ",", RowBox[List[FractionBox["3", "2"], "+", "k"]], ",", FractionBox["3", "2"], ",", RowBox[List["-", "1"]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "k"]]]]], ")"]], "2"]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", RowBox[List["z", "-", "1"]], "]"]], "<", "2"]]]]]]








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> sech </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#10869; </mo> <mrow> <mrow> <mo> - </mo> <mn> 4 </mn> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, FractionBox[&quot;1&quot;, &quot;2&quot;], &quot;)&quot;]], &quot;k&quot;], Pochhammer] </annotation> </semantics> </mrow> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> , </mo> <mrow> <mi> k </mi> <mo> + </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> ; </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> <mo> ; </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;2&quot;, TraditionalForm]], SubscriptBox[&quot;F&quot;, FormBox[&quot;1&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[&quot;1&quot;, &quot;2&quot;], Hypergeometric2F1, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;k&quot;, &quot;+&quot;, FractionBox[&quot;3&quot;, &quot;2&quot;]]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[TagBox[FractionBox[&quot;3&quot;, &quot;2&quot;], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], &quot;;&quot;, TagBox[RowBox[List[&quot;-&quot;, &quot;1&quot;]], Hypergeometric2F1, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] </annotation> </semantics> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &lt; </mo> <mn> 2 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <power /> <apply> <arcsech /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -4 </cn> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <ci> Pochhammer </ci> <cn type='rational'> 1 <sep /> 2 </cn> <ci> k </ci> </apply> <apply> <power /> <apply> <factorial /> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Hypergeometric2F1 </ci> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <ci> k </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <cn type='rational'> 3 <sep /> 2 </cn> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <ci> k </ci> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <lt /> <apply> <abs /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", SuperscriptBox[RowBox[List["ArcSech", "[", "z_", "]"]], "2"], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox["1", "2"], ",", RowBox[List[FractionBox["3", "2"], "+", "k"]], ",", FractionBox["3", "2"], ",", RowBox[List["-", "1"]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "k"]]], RowBox[List["k", "!"]]]]], ")"]], "2"]]], "/;", RowBox[List[RowBox[List["Abs", "[", RowBox[List["z", "-", "1"]], "]"]], "<", "2"]]]]]]]]


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





2007-05-02