0
$\begingroup$

By Euler's formula, I can express $i$ in the following way: $$ i = \cos(\pi/2) + i \sin(\pi/2) = \exp(i \pi/2) $$ I wonder if it is legitimate to write

$$\begin{align}i \sin x &= \exp(i\pi/2) \cdot \frac{\exp(ix)-\exp(-ix)}{2i} \\ &= \frac{\exp(ix)\exp(i\pi/2)-\exp(-ix)\exp(i\pi/2)}{2i} \\ &= \frac{\exp(i(x+\pi/2))-\exp(-i(x+\pi/2))}{2i} \\ &= \sin(x+\pi/2) \end{align}$$

I don't feel like this is right, because it would imply a lot of weird things. So where is my mistake? note: I'm really missing sleep, so please be kind with me.

$\endgroup$
1
$\begingroup$

In your sleep-deprived delirium, you have said $$ e^{-ix}e^{i\pi/2} = e^{-i(x+\pi/2)}$$ where it is actually $$ e^{-ix}e^{i\pi/2} = e^{-i(x-\pi/2)}.$$

It would certainly be strange if $i\sin(x) = \sin(x+\pi/2)$ since one is pure imaginary and the other real for real $x.$

$\endgroup$
  • 1
    $\begingroup$ Thank you, I shouldn't even be trying. I will drown in self-pity till I fall asleep. $\endgroup$ – Marc Oct 26 '17 at 23:30

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.