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Show that the derivative of the complex logarithm of a z belonging to C is edale at 1/z

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Answer to a math question Show that the derivative of the complex logarithm of a z belonging to C is edale at 1/z

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Hermann
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Soit z \in \mathbb{C} . On sait que la fonction logarithme complexe est définie par:

\log(z) = \ln|z| + i \arg(z)

\ln|z| est le logarithme népérien du module de z et \arg(z) est l'argument de z .

Calculons maintenant la dérivée de \log(z) par rapport à z :

\frac{d}{dz}(\log(z)) = \frac{d}{dz}(\ln|z|) + i \frac{d}{dz}(\arg(z))

Puisque \ln|z| ne dépend que du module de z et que \arg(z) ne dépend que de l'argument de z , les dérivées par rapport à z de ces deux parties sont nulles. Ainsi, on obtient :

\frac{d}{dz}(\log(z)) = 0 + i \cdot 0 = 0

Donc la dérivée du logarithme complexe de z par rapport à z est nulle.

\textbf{Réponse:} \frac{d}{dz}(\log(z)) = 0

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