# possible values of imaginary numbers

• Dec 1st 2006, 08:41 AM
markholden
possible values of imaginary numbers
Two similar questions concerning imaginary numbers. What are all the possible values of $$\log(1+i)$$ and of $i^{i}$? Please help, due soon.
• Dec 1st 2006, 09:09 AM
ThePerfectHacker
Quote:

Originally Posted by markholden
Two similar questions concerning imaginary numbers. What are all the possible values of $$\log(1+i)$$ and of $i^{i}$? Please help, due soon.

$i^i=e^{-\pi/2}$
• Dec 1st 2006, 09:14 AM
CaptainBlack
Quote:

Originally Posted by markholden
Two similar questions concerning imaginary numbers. What are all the possible values of $\log(1+i)$ and of $i^{i}$? Please help, due soon.

Suppose $y=\log(1+i)$, by this we mean that:

$e^y=1+i$,

Now suppose $y=u+iv$, then:

$e^u e^{iv}=1+i$, and $|e^u e^{iv}|=e^u=|1+i|=\sqrt{2}$,

so $u=\log(\sqrt{2})$.

Now:

$e^{iv}=\cos(v)+i \sin(v)=1/\sqrt{2}+i/\sqrt{2}$,

so:

$
v=\pi/4 + 2 \pi n,\ \ \ n=0,\ \pm 1,\ \pm 2,\ \ ...
$

Hence:

$
y=\log (\sqrt{2})+\pi i (1/4 +2n),\ \ n=0,\ \pm 1,\ \pm 2,\ \ ...
$

RonL
• Dec 1st 2006, 09:19 AM
CaptainBlack
Quote:

Originally Posted by ThePerfectHacker
$i^i=e^{-\pi/2}$

$
i=e^{i \pi /2 + i 2 \pi n}, \ \ \ n=0, \pm 1,\ \pm 2, ..
$

so:

$
i^i=e^{- \pi /2 - 2 \pi n}, \ \ \ n=0, \pm 1,\ \pm 2, ..
$

RonL