How to solve $e^{ix} = i$

Question

I am taking an on-line course and the following homework problem was posed: $$e^{ix} = i$$ I have no idea how to solve this problem. I have never dealt with solving equations that have imaginary parts. What are the steps to solving such equations? I am familiar with Taylor series and the Euler formula if that is any help.

Thanks.

Answer 1

So, you say that you're okay with Euler's formula: $$e^{ix}=\cos(x)+i\sin(x),$$ and it should be clear that (for real numbers $a,b,c,d$), we have $$a+bi=c+di\quad\iff\quad a=c\;\text{ and }\;b=d.$$ So, which values of $x$ will satisfy $$\cos(x)+\sin(x)i=0+1i\quad ?$$

Answer 2

Well, the geometric meaning of $e^{ix}$ is the weapon here to use.

This is nothing but the point on the complex plane which has length $1$ and angle $x$ measured from the right half of the real axis, in radian. So that $e^{i\pi}=-1$, for example.

Answer 3

$$e^{ix} = i$$ Euler's formula: $$e^{ix}=\cos(x)+i\sin(x),$$ so: $$\cos x+i\sin x = 0+1\cdot i$$ compare real and imaginary parts $$\sin x =1$$ and $$\cos x =0$$ $$x=\dfrac{(4n+1)\pi}{2}\;\;,n \in \mathbb W$$ (W stands for set of whole number $W=${$0,1,2,3,.......,n$}).

Answer 4

Hint:

$$e^{xi}=i=e^{\frac\pi2i+2k\pi i}\implies x=\frac\pi2+2k\pi\;,\;\;k\in\Bbb Z$$

Answer 5

$$ e^{ix} = i \implies ix =\ln(i) \implies ix= \ln(|i|)-i \left(\frac{\pi}{2}+2k\pi\right) $$

$$ \implies x = \frac{\pi}{2}+2k\pi,\quad k\in \mathbb{Z}. $$

Answer 6

A completely different approach:

$$ e^{i\pi} = -1 \implies \sqrt {e^{i\pi}} = \sqrt {-1} \implies e^{\frac{i\pi}{2}} = i $$

Answer 7

The key here is to think in terms of radial co-ordinates rather than cartesian. Whenever you see a complex exponential it's best to think of the geometric interpretation first, if possible.

The radial representation of a complex number $z = a + bi$ is $Re^{i\theta}$
where $R = |z|$ and $\theta=arctan(b/a)$.

So for the number i we ask ourselves what is R, and what is $\theta?$
We look where the point i is on the complex plane - it's on the y axis 1 unit away from the origin. So R is 1 and $\theta$ is $\frac{\pi}{2}$. (Note that you can add any multiple of $2\pi$ to $\theta$ and things stay the same.)

So plug these R, $\theta$ values into the radial form of z to get the radial representation for i which is $1.e^{i\frac{\pi}{2}} = e^{i\frac{\pi}{2}}$
So $e^{ix} = i = e^{i\frac{\pi}{2}}$ and a naive answer is $x = \frac{\pi}{2}$. But remember we could add any multiple of $2{\pi}$ to our angle for i so we need to add $2n{\pi}$ for the general answer.

Hence $x = \frac{\pi}{2}+2n\pi \quad n\in \mathbb{Z}$