inkyvoyd
  • inkyvoyd
e^(ix) = i sin x+cos x. Prove it. Besides using a first order differential equation, and the Maclaurin/Taylor series.
Mathematics
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katieb
  • katieb
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anonymous
  • anonymous
then i guess it depends on your definiation of these functions. i don't think i have ever seen an explanation without power series, because that is how one usually defines \[e^x\]
anonymous
  • anonymous
Is this a homework question?
anonymous
  • anonymous
@inkyvoyd ?

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inkyvoyd
  • inkyvoyd
Nope
anonymous
  • anonymous
Then follow satellite... probably
amistre64
  • amistre64
can we use complex numbers? :)
inkyvoyd
  • inkyvoyd
Sure, just don't go TOO hard.
inkyvoyd
  • inkyvoyd
"i don't think i have ever seen an explanation without power series, because that is how one usually defines" You can prove that the derivatives of e^(ix) and i sin x+cos x are equivalent, which is one of the 2 ways I know how to prove it.
amistre64
  • amistre64
i just seen a rather neat limit type proof
inkyvoyd
  • inkyvoyd
SHOW ME NAO PWEASE
amistre64
  • amistre64
\[\lim_{n\to inf}(1+(z/n))^n=e^z\] where z = a+bi http://en.wikipedia.org/wiki/Euler's_formula
inkyvoyd
  • inkyvoyd
HOW DID I MISS THAT
amistre64
  • amistre64
i just seen it for the first time meself :)
inkyvoyd
  • inkyvoyd
Bumped from last time; wondering if anyone else has proofs
amistre64
  • amistre64
i see that :) i just watched a nice youtube presentation on this where he reinvented the series instead of simply applying hte series; but would that be cheating?
amistre64
  • amistre64
it was something to do with rediscovering the euler formula
inkyvoyd
  • inkyvoyd
But but but, everyone understands the series!
amistre64
  • amistre64
the question pretty much reads: prove that 1+1 = 2 without using anything thats been used prior to prove it. :)
inkyvoyd
  • inkyvoyd
I know, but there are many ways to prove that 1+1 is 2!
amistre64
  • amistre64
not without reinventing the whell there isnt
amistre64
  • amistre64
theres another synapse ill never get back lol
amistre64
  • amistre64
http://www.youtube.com/watch?v=N4ceWhmXxcs i thought this was pretty good, downed a whole bag of microwave popcorn on it last night :)

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