Natural logarithm: Differentiation (formula 01.04.20.0017)

Log

$Mathematica Notation$

Elementary Functions

Log[z]

Differentiation

Fractional integro-differentiation

http://functions.wolfram.com/01.04.20.0017.01

Input Form

D[Log[z]^3, {z, \[Alpha]}] == Piecewise[{{((-1)^\[Alpha] Gamma[\[Alpha]] (Pi^2/2 - 3 (-EulerGamma + Log[z] - PolyGamma[\[Alpha]])^2 - 3 PolyGamma[1, \[Alpha]]))/z^\[Alpha], Element[\[Alpha], Integers] && \[Alpha] > 0}}, (1/(z^\[Alpha] Gamma[1 - \[Alpha]])) (Log[z]^3 + 3 Log[z]^2 (-EulerGamma - PolyGamma[1 - \[Alpha]]) + 3 Log[z] (EulerGamma^2 + Pi^2/6 + 2 EulerGamma PolyGamma[1 - \[Alpha]] + PolyGamma[1 - \[Alpha]]^2 - PolyGamma[1, 1 - \[Alpha]]) - EulerGamma^3 - (EulerGamma Pi^2)/2 - 3 EulerGamma PolyGamma[1 - \[Alpha]]^2 - PolyGamma[1 - \[Alpha]]^3 - (1/2) PolyGamma[1 - \[Alpha]] (6 EulerGamma^2 + Pi^2 - 6 PolyGamma[1, 1 - \[Alpha]]) + 3 EulerGamma PolyGamma[1, 1 - \[Alpha]] + PolyGamma[2, 1] - PolyGamma[2, 1 - \[Alpha]])]

Standard Form

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MathML Form

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</apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <cn type='integer'> 2 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <eulergamma /> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </otherwise> </piecewise> </apply> </annotation-xml> </semantics> </math>

Rule Form

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

2007-05-02

Log[a,z]