Complex number in exponential form

Oct 2006
206
16
I have one question here regarding complex number in exponential form,

Is the following statement true?

\(\displaystyle e^{i\frac{-\pi}{2}}=e^{i\frac{3\pi}{2}}\)

Some ppl say that their power series are not equal, so the statement is wrong.

But to me, it seems like they are equal, right?

Anyone can clarify my doubt? Thanks!!!
 
Oct 2009
4,261
1,836
I have one question here regarding complex number in exponential form,

Is the following statement true?

\(\displaystyle e^{i\frac{-\pi}{2}}=e^{i\frac{3\pi}{2}}\)

Some ppl say that their power series are not equal, so the statement is wrong.

But to me, it seems like they are equal, right?

Anyone can clarify my doubt? Thanks!!!

Of course they're equal: for real \(\displaystyle \phi\,,\,\theta\,,\,\,e^{i\phi}=e^{i\theta}\iff \phi -\theta=2k\pi\,,\,k\in\mathbb{Z}\).

This follows at once from the definition of \(\displaystyle e^{ix}\,,\,x\in\mathbb{R}\) , and the periodicity of the trigonometric functions sine and cosine.

Tonio