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variants of this functions
Factorial






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Factorial[n] > Differentiation > Symbolic differentiation





http://functions.wolfram.com/06.01.20.0004.01









  


  










Input Form





D[n!, {n, m}] == Integrate[(t^n Log[t]^m)/E^t, {t, 1, Infinity}] + (Pochhammer[-m, m]/(n + 1)^(m + 1)) HypergeometricPFQ[ {Subscript[z, 1], Subscript[z, 2], \[Ellipsis], Subscript[z, m]}, {1 + Subscript[z, 1], 1 + Subscript[z, 2], \[Ellipsis], 1 + Subscript[z, m]}, -1] /; Subscript[z, 1] == Subscript[z, 2] == \[Ellipsis] == Subscript[z, m] == n + 1 && Element[m, Integers] && m > 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> <mfrac> <mrow> <msup> <mo> &#8706; </mo> <mi> m </mi> </msup> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> </mrow> <mrow> <mo> &#8706; </mo> <msup> <mi> n </mi> <mi> m </mi> </msup> </mrow> </mfrac> <mo> &#10869; </mo> <mrow> <mrow> <msubsup> <mo> &#8747; </mo> <mn> 1 </mn> <mi> &#8734; </mi> </msubsup> <mrow> <mrow> <msup> <mi> t </mi> <mi> n </mi> </msup> <mo> &#8290; </mo> <mrow> <msup> <mi> log </mi> <mi> m </mi> </msup> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mo> - </mo> <mi> t </mi> </mrow> </msup> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> t </mi> </mrow> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <mtext> </mtext> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mi> m </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;-&quot;, &quot;m&quot;]], &quot;)&quot;]], &quot;m&quot;], Pochhammer] </annotation> </semantics> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> </mfrac> <mo> &#8290; 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</mo> <mrow> <mrow> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> &#10869; </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <mo> &#10869; </mo> <mo> &#8230; </mo> <mo> &#10869; </mo> <msub> <mi> z </mi> <mi> m </mi> </msub> <mo> &#10869; </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mi> m </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> n </ci> <degree> <ci> m </ci> </degree> </bvar> <apply> <factorial /> <ci> n </ci> </apply> </apply> <apply> <plus /> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <ci> t </ci> <ci> n </ci> </apply> <apply> <power /> <apply> <ln /> <ci> t </ci> </apply> <ci> m </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> t </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <ci> Pochhammer </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <ci> m </ci> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <ci> &#8230; 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Rule Form





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





2001-10-29





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