Natural logarithm: Differentiation (formula 01.04.20.0023)

Log

$Mathematica Notation$

Elementary Functions

Log[z]

Differentiation

Fractional integro-differentiation

http://functions.wolfram.com/01.04.20.0023.01

Input Form

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

Standard Form

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MathML Form

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</mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> α </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mi> a </mi> <mo> - </mo> <mi> α </mi> </mrow> </msup> <mtext> </mtext> </mrow> </mtd> <mtd> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> </mtd> </mtr> <mtr> <mtd> <mfrac> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> 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Date Added to functions.wolfram.com (modification date)

2007-05-02

Log[a,z]