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asnaseer
Group Title
What is the shortest way to prove:\[
e^{ix}=\cos(x)+i\sin(x)
\]
 2 years ago
 2 years ago
asnaseer Group Title
What is the shortest way to prove:\[ e^{ix}=\cos(x)+i\sin(x) \]
 2 years ago
 2 years ago

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asnaseer Group TitleBest ResponseYou've already chosen the best response.1
I know one way is to use the series expansion for cos(x) and sin(x) and show that it matches the series expansion for e^(ix)  but is there a shorter proof?
 2 years ago

FoolForMath Group TitleBest ResponseYou've already chosen the best response.1
Do you know what is the polar form of a complex number ?
 2 years ago

Mr.Math Group TitleBest ResponseYou've already chosen the best response.0
I read once a short proof by using differential equations.
 2 years ago

asnaseer Group TitleBest ResponseYou've already chosen the best response.1
do you recall that proof @Mr.Math?
 2 years ago

Mr.Math Group TitleBest ResponseYou've already chosen the best response.0
Look here http://en.wikipedia.org/wiki/Euler%27s_formula#Proofs
 2 years ago

asnaseer Group TitleBest ResponseYou've already chosen the best response.1
@FoolForMath  do you mean this: dw:1325522941156:dw
 2 years ago

asnaseer Group TitleBest ResponseYou've already chosen the best response.1
thanks @Mr.Math  that is what I was looking for.
 2 years ago

Mr.Math Group TitleBest ResponseYou've already chosen the best response.0
You're welcome!
 2 years ago

FoolForMath Group TitleBest ResponseYou've already chosen the best response.1
asnaseer take a look at this thread: http://math.stackexchange.com/questions/3510/howtoproveeulersformulaexpitcostisint
 2 years ago

asnaseer Group TitleBest ResponseYou've already chosen the best response.1
thanks @FoolForMath
 2 years ago

FoolForMath Group TitleBest ResponseYou've already chosen the best response.1
The complex number approach has been explained there too.
 2 years ago

satellite73 Group TitleBest ResponseYou've already chosen the best response.0
i don't think there is a shorter way than series expansion
 2 years ago

FoolForMath Group TitleBest ResponseYou've already chosen the best response.1
@asnaseer: Glad to help :)
 2 years ago

asnaseer Group TitleBest ResponseYou've already chosen the best response.1
Using the uniqueness theorem with differentials seems to be the shortest method.
 2 years ago

asnaseer Group TitleBest ResponseYou've already chosen the best response.1
I guess it all depends on how you define "shortest"
 2 years ago

JamesJ Group TitleBest ResponseYou've already chosen the best response.1
...and elementary. Without looking at all the alternatives in a lot of detail, I'd hypothesize the series proof is the most mathematically elementary.
 2 years ago

asnaseer Group TitleBest ResponseYou've already chosen the best response.1
I still find the proof using derivatives much simpler.
 2 years ago

FoolForMath Group TitleBest ResponseYou've already chosen the best response.1
Yes, I agree with you asnaseer.
 2 years ago

asnaseer Group TitleBest ResponseYou've already chosen the best response.1
it look more "elegant" as well.
 2 years ago

FoolForMath Group TitleBest ResponseYou've already chosen the best response.1
yea but inaccessible without knowledge of calculus.
 2 years ago

JamesJ Group TitleBest ResponseYou've already chosen the best response.1
Returning for a moment to the idea of being elementary, what the differentiation proofs assume is that e^ix is differentiable. That's not obvious before the fact.
 2 years ago

FoolForMath Group TitleBest ResponseYou've already chosen the best response.1
btw asnaseer, what's your need?
 2 years ago

asnaseer Group TitleBest ResponseYou've already chosen the best response.1
no need really  I was just wondering if there were any other ways to prove this apart from the series expansion.
 2 years ago
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