Question

e^(ix) = i sin x+cos x. Prove it. Besides using a first order differential equation, and the Maclaurin/Taylor series.

Answer 1

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
\[e^x\]

Answer 2

Is this a homework question?

Answer 3

@inkyvoyd ?

Answer 4

Nope

Answer 5

Then follow satellite... probably

Answer 6

can we use complex numbers? :)

Answer 7

Sure, just don't go TOO hard.

Answer 8

"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.

Answer 9

i just seen a rather neat limit type proof

Answer 10

SHOW ME NAO PWEASE

Answer 11

\[\lim_{n\to inf}(1+(z/n))^n=e^z\]
where z = a+bi

http://en.wikipedia.org/wiki/Euler's_formula

Answer 12

HOW DID I MISS THAT

Answer 13

i just seen it for the first time meself :)

Answer 14

Bumped from last time; wondering if anyone else has proofs

Answer 15

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?

Answer 16

it was something to do with rediscovering the euler formula

Answer 17

But but but, everyone understands the series!

Answer 18

the question pretty much reads: prove that 1+1 = 2 without using anything thats been used prior to prove it. :)

Answer 19

I know, but there are many ways to prove that 1+1 is 2!

Answer 20

not without reinventing the whell there isnt

Answer 21

theres another synapse ill never get back lol

Answer 22

http://www.youtube.com/watch?v=N4ceWhmXxcs

i thought this was pretty good, downed a whole bag of microwave popcorn on it last night :)