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variants of this functions
Log






Mathematica Notation

Traditional Notation









Elementary Functions > Log[z] > Differentiation > Fractional integro-differentiation





http://functions.wolfram.com/01.04.20.0025.01









  


  










Input Form





D[Log[f[z]], {z, \[Alpha]}] == (2 I Pi Floor[(Pi - Arg[f[0]] - Arg[f[z]/f[0]])/(2 Pi)] + Log[f[0]]) (1/(z^\[Alpha] Gamma[1 - \[Alpha]])) + Sum[((-1)^k/(k + 1)) (Derivative[u][f][0]/(f[0] u!))^(k + 1) Subscript[p, k + 1, s - u k] (Gamma[s + u + 1]/ Gamma[s + u - \[Alpha] + 1]) z^(s + u - \[Alpha]), {s, 0, Infinity}, {k, 0, s/u}] /; f[0] != 0 && (Derivative[k][f][0] == 0 /; 1 <= k <= u - 1) && Derivative[u][f][0] != 0 && Subscript[p, j, 0] == 1 && Subscript[p, j, k] == (u!/(Derivative[u][f][0] k)) Sum[((j m - k + m)/((m + u)! (m + 1)!)) Derivative[m + u][f][0] Subscript[p, j, k - m], {m, 1, k}] && Element[k, Integers] && k > 0










Standard Form





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MathML Form







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</mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> s </mi> <mo> + </mo> <mi> u </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> s </mi> <mo> + </mo> <mi> u </mi> <mo> - </mo> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mrow> <msup> <mi> f </mi> <semantics> <mrow> <mo> ( </mo> <mi> u </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, &quot;u&quot;, &quot;)&quot;]], Derivative] </annotation> </semantics> </msup> <mo> ( </mo> <mn> 0 </mn> <mo> ) </mo> </mrow> <mrow> <mrow> <mi> f </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 0 </mn> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> u </mi> <mo> ! </mo> </mrow> </mrow> </mfrac> <mo> ) </mo> </mrow> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <msub> <mi> p </mi> <mrow> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mi> s </mi> <mo> - </mo> <mrow> <mi> u </mi> <mo> &#8290; </mo> <mi> k </mi> </mrow> </mrow> </mrow> </msub> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mi> s </mi> <mo> + </mo> <mi> u </mi> <mo> - </mo> <mi> &#945; </mi> </mrow> </msup> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> f </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 0 </mn> <mo> ) </mo> </mrow> <mo> &#8800; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <msup> <mi> f </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> 0 </mn> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mn> 0 </mn> </mrow> <mo> /; </mo> <mrow> <mn> 1 </mn> <mo> &#8804; </mo> <mi> k </mi> <mo> &#8804; </mo> <mrow> <mi> u </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <msup> <mi> f </mi> <semantics> <mrow> <mo> ( </mo> <mi> u </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, &quot;u&quot;, &quot;)&quot;]], Derivative] </annotation> </semantics> </msup> <mo> ( </mo> <mn> 0 </mn> <mo> ) </mo> </mrow> <mo> &#8800; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> p </mi> <mrow> <mi> j </mi> <mo> , </mo> <mn> 0 </mn> </mrow> </msub> <mo> &#10869; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> p </mi> <mrow> <mi> j </mi> <mo> , </mo> <mi> k </mi> </mrow> </msub> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <mrow> <mi> u </mi> <mo> ! </mo> </mrow> <mtext> </mtext> </mrow> <mrow> <mrow> <msup> <mi> f </mi> <semantics> <mrow> <mo> ( </mo> <mi> u </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, &quot;u&quot;, &quot;)&quot;]], Derivative] </annotation> </semantics> </msup> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> k </mi> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> m </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> k </mi> </munderover> <mrow> <mfrac> <mrow> <mrow> <mi> j </mi> <mo> &#8290; </mo> <mi> m </mi> </mrow> <mo> + </mo> <mi> m </mi> <mo> - </mo> <mi> k </mi> </mrow> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mi> u </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msup> <mi> f </mi> <semantics> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mi> u </mi> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;m&quot;, &quot;+&quot;, &quot;u&quot;]], &quot;)&quot;]], Derivative] </annotation> </semantics> </msup> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msub> <mi> p </mi> <mrow> <mi> j </mi> <mo> , </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> m </mi> </mrow> </mrow> </msub> </mrow> </mrow> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mi> k </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> z </ci> <degree> <ci> &#945; </ci> </degree> </bvar> <apply> <ln /> <apply> <ci> f </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <pi /> <apply> <floor /> <apply> <times /> <apply> <plus /> <pi /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <arg /> <apply> <times /> <apply> <ci> f </ci> <ci> z </ci> </apply> <apply> <power /> <apply> <ci> f </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <arg /> <apply> <ci> f </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <ln /> <apply> <ci> f </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; 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</ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <ci> D </ci> <apply> <ci> f </ci> <cn type='integer'> 0 </cn> </apply> <list> <cn type='integer'> 0 </cn> <ci> u </ci> </list> </apply> <apply> <power /> <apply> <times /> <apply> <ci> f </ci> <cn type='integer'> 0 </cn> </apply> <apply> <factorial /> <ci> u </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> p </ci> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> s </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> u </ci> <ci> k </ci> </apply> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <ci> s </ci> <ci> u </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <neq /> <apply> <ci> f </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> D </ci> <apply> <ci> f </ci> <cn type='integer'> 0 </cn> </apply> <list> <cn type='integer'> 0 </cn> <ci> k </ci> </list> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <leq /> <cn type='integer'> 1 </cn> <ci> k </ci> <apply> <plus /> <ci> u </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <neq /> <apply> <ci> D </ci> <apply> <ci> f </ci> <cn type='integer'> 0 </cn> </apply> <list> <cn type='integer'> 0 </cn> <ci> u </ci> </list> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> p </ci> <ci> j </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> p </ci> <ci> j </ci> <ci> k </ci> </apply> <apply> <times /> <apply> <times /> <apply> <factorial /> <ci> u </ci> </apply> <apply> <power /> <apply> <times /> <apply> <ci> D </ci> <apply> <ci> f </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <list> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <ci> u </ci> </list> </apply> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> m </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> k </ci> </uplimit> <apply> <times /> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> j </ci> <ci> m </ci> </apply> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <apply> <plus /> <ci> m </ci> <ci> u </ci> </apply> </apply> <apply> <factorial /> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> D </ci> <apply> <ci> f </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <list> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <apply> <plus /> <ci> m </ci> <ci> u </ci> </apply> </list> </apply> <apply> <ci> Subscript </ci> <ci> p </ci> <ci> j </ci> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> k </ci> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2007-05-02





© 1998- Wolfram Research, Inc.