Natural logarithm: Differentiation (formula 01.04.20.0018)

Log

$Mathematica Notation$

Elementary Functions

Log[z]

Differentiation

Fractional integro-differentiation

http://functions.wolfram.com/01.04.20.0018.01

Input Form

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

Standard Form

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MathML Form

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Rule Form

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

2007-05-02

Log[a,z]