$$e^{ix} = i$$

Question

$$e^{ix} = i$$

jtvatsim · Answer

$$x = \pi/2$$

unklerhaukus · Answer

that's one solution, @jtvatsim .

But what is the general solution?

jtvatsim · Answer

lol, of course, $$x = \pi/2 + 2\pi k, \  k \in \mathbb{Z}$$

jhannybean · Answer

This looks like something @wio taught me a few days ago.

unklerhaukus · Answer

that's better!

unklerhaukus · Answer

Medal goes to full working.

unklerhaukus · Answer

try: Euler's formula

solomonzelman · Answer

$\LARGE\color{blue}{  e^{i x} =\cos(x) +i\sin(x)  }$
So that means,
$\LARGE\color{blue}{  i =\cos(x) +i\sin(x)  }$

unklerhaukus · Answer

that's a good start, 

now compare the real and imaginary components on both sides

unklerhaukus · Answer

i.e.    a + ib   =  c + id
  
==>    a = c    and   b = d

anonymous · Answer

Wouldn't general solution be: $$
\pi/2+2\pi n
$$

unklerhaukus · Answer

@wio, yes that is equivalent to @jtvatsim's (second) answer
where   n = k = some integer

anonymous · Answer

What exactly are you looking for then, in terms of work?  
$1=\sin(x)\implies x=\pi/2+2\pi n$?

unklerhaukus · Answer

$$\begin{align}
				e^{ix}	&= i								\
			\cos x +i\sin x	&= i								\
				\sin x 	&= 1							    	\
					x	&= \arcsin(1)					    	\
						&= \frac\pi2+2n\pi	&& n\in\mathbb Z	\
					%	&= \frac{1+4n}2\pi
		\end{align}$$