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Power






Mathematica Notation

Traditional Notation









Elementary Functions > Power[z,a] > Series representations > Generalized power series > Expansions of (1+z)a at z==0 > Expansions of (1+Sumk=1infinityck zk)a at z==0





http://functions.wolfram.com/01.02.06.0061.01









  


  










Input Form





(1 + Sum[Subscript[c, k] z^k, {k, 1, Infinity}])^a \[Proportional] 1 + a Subscript[c, 1] z + (1/2) a ((-1 + a) Subscript[c, 1]^2 + 2 Subscript[c, 2]) z^2 + (1/6) a ((a - 1) (a - 2) Subscript[c, 1]^3 + 6 (a - 1) Subscript[c, 1] Subscript[c, 2] + 6 Subscript[c, 3]) z^3 + O[z^4] /; (z -> 0)










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> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <msub> <mi> c </mi> <mi> k </mi> </msub> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> k </mi> </msup> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mi> a </mi> </msup> <mo> &#8733; </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <msub> <mi> c </mi> <mn> 1 </mn> </msub> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mi> a </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msubsup> <mi> c </mi> <mn> 1 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msub> <mi> c </mi> <mn> 2 </mn> </msub> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 6 </mn> </mfrac> <mo> &#8290; </mo> <mi> a </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msubsup> <mi> c </mi> <mn> 1 </mn> <mn> 3 </mn> </msubsup> </mrow> <mo> + </mo> <mrow> <mn> 6 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msub> <mi> c </mi> <mn> 2 </mn> </msub> <mo> &#8290; </mo> <msub> <mi> c </mi> <mn> 1 </mn> </msub> </mrow> <mo> + </mo> <mrow> <mn> 6 </mn> <mo> &#8290; </mo> <msub> <mi> c </mi> <mn> 3 </mn> </msub> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mi> O </mi> <mo> &#8289; </mo> <mo> ( </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <ci> Subscript </ci> <ci> c </ci> <ci> k </ci> </apply> <apply> <power /> <ci> z </ci> <ci> k </ci> </apply> </apply> </apply> </apply> <ci> a </ci> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <ci> a </ci> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 1 </cn> </apply> <ci> z </ci> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <ci> a </ci> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 6 </cn> <ci> a </ci> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> a </ci> <cn type='integer'> -2 </cn> </apply> <apply> <plus /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 6 </cn> <apply> <plus /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 6 </cn> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <ci> O </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> </apply> <apply> <ci> Rule </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2007-05-02





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