EulerE[n, z] \[Proportional] EulerE[n, Subscript[z, 0]] + n EulerE[-1 + n, Subscript[z, 0]] (z - Subscript[z, 0]) + (1/2) (-1 + n) n EulerE[-2 + n, Subscript[z, 0]] (z - Subscript[z, 0])^2 + O[(z - Subscript[z, 0])^3]

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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> <msub> <semantics> <mi> E </mi> <annotation encoding='Mathematica'> TagBox[&quot;E&quot;, EulerE] </annotation> </semantics> <mi> n </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8733; </mo> <mrow> <mrow> <msub> <semantics> <mi> E </mi> <annotation encoding='Mathematica'> TagBox[&quot;E&quot;, EulerE] </annotation> </semantics> <mi> n </mi> </msub> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> n </mi> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> E </mi> <annotation encoding='Mathematica'> TagBox[&quot;E&quot;, EulerE] </annotation> </semantics> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> n </mi> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> E </mi> <annotation encoding='Mathematica'> TagBox[&quot;E&quot;, EulerE] </annotation> </semantics> <mrow> <mi> n </mi> <mo> - </mo> <mn> 2 </mn> </mrow> </msub> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mi> O </mi> <mo> &#8289; </mo> <mo> ( </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Proportional </ci> <apply> <ci> EulerE </ci> <ci> n </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <ci> EulerE </ci> <ci> n </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <times /> <ci> n </ci> <apply> <ci> EulerE </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <ci> n </ci> <apply> <ci> EulerE </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ci> O </ci> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2007-05-02