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variants of this functions
Gamma




Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Gamma[z] > Differentiation > Fractional integro-differentiation





http://functions.wolfram.com/06.05.20.0005.01









  


  








Input Form





D[Gamma[z], {z, \[Alpha]}] == z^(-1 - \[Alpha]) FDPowerConstant[z, -1, \[Alpha]] + Sum[(Derivative[k][Gamma][1]/(k Gamma[k - \[Alpha]])) z^(k - \[Alpha] - 1), {k, 1, Infinity}]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "\[Alpha]"]], "}"]]], RowBox[List["Gamma", "[", "z", "]"]]]], "\[Equal]", RowBox[List[RowBox[List[SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "-", "\[Alpha]"]]], " ", RowBox[List["FDPowerConstant", "[", RowBox[List["z", ",", RowBox[List["-", "1"]], ",", "\[Alpha]"]], "]"]]]], "+", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox["Gamma", TagBox[RowBox[List["(", "k", ")"]], Derivative], Rule[MultilineFunction, None]], "[", "1", "]"]], RowBox[List["k", " ", RowBox[List["Gamma", "[", RowBox[List["k", "-", "\[Alpha]"]], "]"]]]]], SuperscriptBox["z", RowBox[List["k", "-", "\[Alpha]", "-", "1"]]]]]]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mfrac> <mrow> <msup> <mo> &#8706; </mo> <mi> &#945; </mi> </msup> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> &#8706; </mo> <msup> <mi> z </mi> <mi> &#945; </mi> </msup> </mrow> </mfrac> <mo> &#10869; </mo> <mtext> </mtext> <mrow> <mrow> <mrow> <msubsup> <mi> &#8497;&#119966; </mi> <mi> exp </mi> <mrow> <mo> ( </mo> <mi> &#945; </mi> <mo> ) </mo> </mrow> </msubsup> <mo> ( </mo> <mrow> <mi> z </mi> <mo> , </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mrow> <mo> - </mo> <mi> &#945; </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> </mrow> <mo> + </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mfrac> <mrow> <mtext> </mtext> <mrow> <mrow> <msup> <mi> &#915; </mi> <semantics> <mrow> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, &quot;k&quot;, &quot;)&quot;]], Derivative] </annotation> </semantics> </msup> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mi> k </mi> <mo> - </mo> <mi> &#945; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> </mrow> </mrow> <mrow> <mi> k </mi> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> &#945; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> z </ci> <degree> <ci> &#945; </ci> </degree> </bvar> <apply> <ci> Gamma </ci> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> &#8497;&#119966; </ci> <ci> exp </ci> </apply> <ci> &#945; </ci> </apply> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <ci> D </ci> <apply> <ci> Gamma </ci> <cn type='integer'> 1 </cn> </apply> <list> <cn type='integer'> 1 </cn> <ci> k </ci> </list> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <ci> k </ci> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "\[Alpha]_"]], "}"]]]]], RowBox[List["Gamma", "[", "z_", "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "-", "\[Alpha]"]]], " ", RowBox[List["FDPowerConstant", "[", RowBox[List["z", ",", RowBox[List["-", "1"]], ",", "\[Alpha]"]], "]"]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List[SuperscriptBox["Gamma", TagBox[RowBox[List["(", "k", ")"]], Derivative], Rule[MultilineFunction, None]], "[", "1", "]"]], " ", SuperscriptBox["z", RowBox[List["k", "-", "\[Alpha]", "-", "1"]]]]], RowBox[List["k", " ", RowBox[List["Gamma", "[", RowBox[List["k", "-", "\[Alpha]"]], "]"]]]]]]]]]]]]]


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





2001-10-29