Thread: possible values of imaginary numbers

1. possible values of imaginary numbers

Two similar questions concerning imaginary numbers. What are all the possible values of [Math]\log(1+i)[/tex] and of $\displaystyle i^{i}$? Please help, due soon.

2. Originally Posted by markholden
Two similar questions concerning imaginary numbers. What are all the possible values of [Math]\log(1+i)[/tex] and of $\displaystyle i^{i}$? Please help, due soon.
$\displaystyle i^i=e^{-\pi/2}$

3. Originally Posted by markholden
Two similar questions concerning imaginary numbers. What are all the possible values of $\displaystyle \log(1+i)$ and of $\displaystyle i^{i}$? Please help, due soon.
Suppose $\displaystyle y=\log(1+i)$, by this we mean that:

$\displaystyle e^y=1+i$,

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

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

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

Now:

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

so:

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

Hence:

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

RonL

4. Originally Posted by ThePerfectHacker
$\displaystyle i^i=e^{-\pi/2}$
$\displaystyle i=e^{i \pi /2 + i 2 \pi n}, \ \ \ n=0, \pm 1,\ \pm 2, ..$

so:

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

RonL