e^ipi + 1 = 0

For college and university level advanced Mathematics
Dipan
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e^ipi + 1 = 0

Unread post by Dipan » Thu Jan 20, 2011 11:23 pm

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

AntiviruShahriar
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Re: e^ipi + 1 = 0

Unread post by AntiviruShahriar » Fri Jan 21, 2011 2:02 am

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

tanvirab
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Re: e^ipi + 1 = 0

Unread post by tanvirab » Fri Jan 21, 2011 2:36 am

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 $i$).

tanvirab
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Re: e^ipi + 1 = 0

Unread post by tanvirab » Fri Jan 21, 2011 2:38 am


Dipan
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Re: e^ipi + 1 = 0

Unread post by Dipan » Fri Jan 21, 2011 10:10 am

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

tanvirab
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Re: e^ipi + 1 = 0

Unread post by tanvirab » Fri Jan 21, 2011 10:13 am

It's not allowed to talk about divisional olympiad questions

abir91
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Re: e^ipi + 1 = 0

Unread post by abir91 » Fri Jan 21, 2011 4:24 pm

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

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Dipan
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Re: e^ipi + 1 = 0

Unread post by Dipan » Fri Jan 21, 2011 11:12 pm

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

abir91
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Re: e^ipi + 1 = 0

Unread post by abir91 » Fri Jan 21, 2011 11:32 pm

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.
Abir

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tanvirab
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Re: e^ipi + 1 = 0

Unread post by tanvirab » Sat Jan 22, 2011 1:18 am

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.

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