Complex exponentials.

Definition 4.6.1       

For any $z \in \mathbb{C}^*$ and every $\alpha \in \mathbb{C}$

$z^{\alpha} = e^ {\alpha \log z}$

Example 4.6.2       

$16^{\frac 12}$	$= e^{\frac 12 \log 16}$	$= e^{\frac 12 (\ln 16 + 2k \pi i )}$
	$= e^{\frac 12 (4 \ln 2 + 2k \pi i )}$	$= e^{2 \ln 2 + k \pi i }$
	$= e^{2 \ln 2} \cdot e^{k \pi i }$	$= 4 e^{k \pi i }$
	$= \pm 4$ .

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Proposition 4.6.3   $\frac {d}{dz}(\alpha ^z)= \alpha ^z \; \log \alpha$ .

For the proof, we use the definition $\alpha ^z=e^{z \log \alpha}$ . Then we have:

$$\frac {d}{dz}(\alpha ^z)= \frac {d}{dz}(e^{z \log \alpha}) =\log \alpha \; e^{z \log \alpha} = \alpha ^z \; \log \alpha.$$

Proposition 4.6.4   $\frac {d}{dz}(z^{\alpha })= \alpha z^ {\alpha -1}$ .

For the proof, we use the definition $z^{\alpha} = e^ {\alpha \log z}$ . Then we have:

$$\frac {d}{dz}(z^{\alpha })= \frac {d}{dz}(\alpha \log z) \; e^{\a... ...{z} e^{\alpha \log z} = \frac {\alpha }{z} z^ {\alpha } =\alpha z^ {\alpha -1}.$$