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

This Question is Closed

asnaseer
 4 years ago
Best ResponseYou've already chosen the best response.1I 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?

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0Do you know what is the polar form of a complex number ?

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

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

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

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

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

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0asnaseer take a look at this thread: http://math.stackexchange.com/questions/3510/howtoproveeulersformulaexpitcostisint

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0The complex number approach has been explained there too.

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0i don't think there is a shorter way than series expansion

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0@asnaseer: Glad to help :)

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

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

JamesJ
 4 years ago
Best 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.

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

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0Yes, I agree with you asnaseer.

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

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0yea but inaccessible without knowledge of calculus.

JamesJ
 4 years ago
Best ResponseYou've already chosen the best response.1Returning 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.

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0btw asnaseer, what's your need?

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