asnaseer
  • asnaseer
What is the shortest way to prove:\[ e^{ix}=\cos(x)+i\sin(x) \]
Mathematics
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schrodinger
  • schrodinger
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asnaseer
  • asnaseer
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?
anonymous
  • anonymous
Do you know what is the polar form of a complex number ?
asnaseer
  • asnaseer
yes

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Mr.Math
  • Mr.Math
I read once a short proof by using differential equations.
asnaseer
  • asnaseer
do you recall that proof @Mr.Math?
Mr.Math
  • Mr.Math
Look here http://en.wikipedia.org/wiki/Euler%27s_formula#Proofs
asnaseer
  • asnaseer
@FoolForMath - do you mean this: |dw:1325522941156:dw|
asnaseer
  • asnaseer
thanks @Mr.Math - that is what I was looking for.
Mr.Math
  • Mr.Math
You're welcome!
anonymous
  • anonymous
asnaseer take a look at this thread: http://math.stackexchange.com/questions/3510/how-to-prove-eulers-formula-expi-t-costi-sint
asnaseer
  • asnaseer
thanks @FoolForMath
anonymous
  • anonymous
The complex number approach has been explained there too.
anonymous
  • anonymous
i don't think there is a shorter way than series expansion
anonymous
  • anonymous
@asnaseer: Glad to help :)
asnaseer
  • asnaseer
Using the uniqueness theorem with differentials seems to be the shortest method.
asnaseer
  • asnaseer
I guess it all depends on how you define "shortest"
JamesJ
  • JamesJ
...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
  • asnaseer
I still find the proof using derivatives much simpler.
anonymous
  • anonymous
Yes, I agree with you asnaseer.
asnaseer
  • asnaseer
it look more "elegant" as well.
asnaseer
  • asnaseer
*looks
anonymous
  • anonymous
yea but inaccessible without knowledge of calculus.
JamesJ
  • JamesJ
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.
anonymous
  • anonymous
btw asnaseer, what's your need?
asnaseer
  • asnaseer
no need really - I was just wondering if there were any other ways to prove this apart from the series expansion.

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