Exercise [05.10] c

susumu · 02 Jul 2008, 21:15

The ambiguity of complex powers leads us into problems if we are not careful. If w^z=a

by specifying a particular $\log w$ , and if w^z=b

by specifying a different
$\log w$ , then we must not mix them and conclude that a=b

.

For this 'paradox' we have:
$e^{1+2\pi i}=e$ by specifying $\log e=1$

$\left(e^{1+2\pi i}\right)^{1+2\pi i}=e$
by specifying $\log\left(e^{1+2\pi i}\right)=\log e=1$

$\left(e^{1+2\pi i}\right)^{1+2\pi i}=e^{1+4\pi i-4\pi^2}=e^{1-4\pi^2}$
by specifying $\log\left(e^{1+2\pi i}\right)=(1+2\pi i)\log e=1+2\pi i$

The second and third equality above hold under different specifications of $\log\left(e^{1+2\pi i}\right)$ . Therefore, there is no
paradox at all, only we were misled by mixing them in a single equality.

Exercise [05.10] c

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