BdMO Online Forum

In a book I have got that ,
e^i pi = cos pi + i sin pi.
but how?????

"College / University Level" e r 1ta post kora ache..oita poira dekho.....post name "$i^i$"

Do you know about Taylor series? The Taylor series for exponential function can be generalized to complex numbers. Then it's just a matter of writing it as the sum of the Taylor series of cosine and the Taylor series of sine (multiplied by $i$).

Here is the proof : http://en.wikipedia.org/wiki/Euler%27s_ ... sis#Proofs

thanks tanvir bhai...the link is really helpful...
@Anti...tomar divisional mo te ki asche...

It's not allowed to talk about divisional olympiad questions

Note that, dealing with infinity without understanding Analysis properly can lead to nasty disasters. So although the proof is correct, I think one will appreciate the result properly when he can fill the gaps. If you can't, for the time being, just remember that there are things that needs to be proved before you can prove it.

*If you are familiar with analysis, then ignore my comment.

@Abir bhai...Have you told this to me?I can't understand what you wanted to say..Please, make it clear to me..

No. It was intended for a general audience, not necessarily only you.

I meant there are gaps in that proof which needs to proved (the author did not include them is because he assumed the audience knows them), but if you don't see them right now, don't worry, just know that there are gaps and you will prove them later when you learn how to treat these things rigorously.

I think waht Abir is pointing out, that the series in the proof is infinite, and infinite sums needs to be defined and treated very carefully; You have to worry about the right kind of convergence and the results might be completely different for different kind of convergence (although that's very rare). If you do not know what this technicalities mean (which I assume you do not, since these things are taught only in an advanced analysis class at universities), then don't worry about them for now. Keep the general idea in mind, and if you study mathematics when you get to university, you will know the complete picture.