## asnaseer Group Title What is the shortest way to prove:$e^{ix}=\cos(x)+i\sin(x)$ 2 years ago 2 years ago

1. asnaseer Group Title

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?

2. FoolForMath Group Title

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

3. asnaseer Group Title

yes

4. Mr.Math Group Title

I read once a short proof by using differential equations.

5. asnaseer Group Title

do you recall that proof @Mr.Math?

6. Mr.Math Group Title
7. asnaseer Group Title

@FoolForMath - do you mean this: |dw:1325522941156:dw|

8. asnaseer Group Title

thanks @Mr.Math - that is what I was looking for.

9. Mr.Math Group Title

You're welcome!

10. FoolForMath Group Title

asnaseer take a look at this thread: http://math.stackexchange.com/questions/3510/how-to-prove-eulers-formula-expi-t-costi-sint

11. asnaseer Group Title

thanks @FoolForMath

12. FoolForMath Group Title

The complex number approach has been explained there too.

13. satellite73 Group Title

i don't think there is a shorter way than series expansion

14. FoolForMath Group Title

15. asnaseer Group Title

Using the uniqueness theorem with differentials seems to be the shortest method.

16. asnaseer Group Title

I guess it all depends on how you define "shortest"

17. JamesJ Group Title

...and elementary. Without looking at all the alternatives in a lot of detail, I'd hypothesize the series proof is the most mathematically elementary.

18. asnaseer Group Title

I still find the proof using derivatives much simpler.

19. FoolForMath Group Title

Yes, I agree with you asnaseer.

20. asnaseer Group Title

it look more "elegant" as well.

21. asnaseer Group Title

*looks

22. FoolForMath Group Title

yea but inaccessible without knowledge of calculus.

23. JamesJ Group Title

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.

24. FoolForMath Group Title