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Pochhammer




Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Pochhammer[a,n] > Series representations > Generalized power series > Expansions of (-m+Epsilon)n at Epsilon==0/;m-n ∉ N && m ∈ N





http://functions.wolfram.com/06.10.06.0019.01









  


  








Input Form





Pochhammer[-m + \[Epsilon], n] \[Proportional] (-1)^m m! Gamma[-m + n] \[Epsilon] (1 + O[\[Epsilon]]) /; !(Element[m - n, Integers] && m - n >= 0) && Element[m, Integers] && m >= 0








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["-", "m"]], "+", "\[Epsilon]"]], ",", "n"]], "]"]], "\[Proportional]", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], " ", RowBox[List["m", "!"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "m"]], "+", "n"]], "]"]], "\[Epsilon]", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", "\[Epsilon]", "]"]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["Not", "[", RowBox[List[RowBox[List[RowBox[List["m", "-", "n"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["m", "-", "n"]], "\[GreaterEqual]", "0"]]]], "]"]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", "\[GreaterEqual]", "0"]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> m </mi> </mrow> <mo> + </mo> <mi> &#1013; </mi> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[RowBox[List[&quot;-&quot;, &quot;m&quot;]], &quot;+&quot;, &quot;\[Epsilon]&quot;]], &quot;)&quot;]], &quot;n&quot;], Pochhammer] </annotation> </semantics> <mo> &#8733; </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> m </mi> </msup> <mo> &#8290; </mo> <mtext> </mtext> <mrow> <mi> m </mi> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#1013; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> O </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#1013; </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> m </mi> <mo> - </mo> <mi> n </mi> </mrow> <mo> &#8713; </mo> <semantics> <mi> &#8469; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalN]&quot;, Function[List[], Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> m </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> Pochhammer </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <ci> &#1013; </ci> </apply> <ci> n </ci> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <apply> <factorial /> <ci> m </ci> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> <ci> &#1013; </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <ci> O </ci> <ci> &#1013; </ci> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <notin /> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <integers /> </apply> <apply> <in /> <ci> m </ci> <ci> &#8469; </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["-", "m_"]], "+", "\[Epsilon]_"]], ",", "n_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], " ", RowBox[List["m", "!"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "m"]], "+", "n"]], "]"]], " ", "\[Epsilon]", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", "\[Epsilon]", "]"]]]], ")"]]]], "/;", RowBox[List[RowBox[List["!", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["m", "-", "n"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["m", "-", "n"]], "\[GreaterEqual]", "0"]]]], ")"]]]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[GreaterEqual]", "0"]]]]]]]]]]


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





2007-05-02