anonymous
  • anonymous
Explain this identity.
Mathematics
chestercat
  • chestercat
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anonymous
  • anonymous
hmm... is that e^(ix) + e^(ix) so = 2e^(ix) on the right side?
anonymous
  • anonymous
\[\cos(x)=\frac{e^{ix}+e^{-ix}}{2} \]
anonymous
  • anonymous
that means I can split cos(x) into two different functions right?

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anonymous
  • anonymous
what do you mean split up cosx ? that looks a bit like the hperbolic coshx...
anonymous
  • anonymous
Well there is a problem where I try to show if cos(x) is an eigenfunction of an operator. I did the operation and found out it wasn't but then my book says if it's not a the eigenfunction that it must be a superposition eigenfunction so it splits it into two functions. I just dont see how it splits or how this superposition will be the eigenfunction.
anonymous
  • anonymous
Can you at least show me how \[\cos(x)=\frac{e^{ix}+e^{-ix}}{2} \]
anonymous
  • anonymous
this is guess, but try doing the series expansion for e^(ix) + e^(-ix). i have a feeling the imaginary terms will cancel.
anonymous
  • anonymous
Write e^ix =cos(x)+i sin(x)
anonymous
  • anonymous
e^(ix) = cos x + i sin x (Euler's Formula) e^(-ix) = cos(-x) + i sin (-x) = cos x- i sin x Add them.

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