Ace school

with brainly

  • Get help from millions of students
  • Learn from experts with step-by-step explanations
  • Level-up by helping others

A community for students.

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

Mathematics
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.

Join Brainly to access

this expert answer

SEE EXPERT ANSWER

To see the expert answer you'll need to create a free account at Brainly

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?
Do you know what is the polar form of a complex number ?
yes

Not the answer you are looking for?

Search for more explanations.

Ask your own question

Other answers:

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

Not the answer you are looking for?

Search for more explanations.

Ask your own question