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asnaseer

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

  • 2 years ago
  • 2 years ago

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  1. asnaseer
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    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 years ago
  2. FoolForMath
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    Do you know what is the polar form of a complex number ?

    • 2 years ago
  3. asnaseer
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    yes

    • 2 years ago
  4. Mr.Math
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    I read once a short proof by using differential equations.

    • 2 years ago
  5. asnaseer
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    do you recall that proof @Mr.Math?

    • 2 years ago
  6. Mr.Math
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    Look here http://en.wikipedia.org/wiki/Euler%27s_formula#Proofs

    • 2 years ago
  7. asnaseer
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    @FoolForMath - do you mean this: |dw:1325522941156:dw|

    • 2 years ago
  8. asnaseer
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    thanks @Mr.Math - that is what I was looking for.

    • 2 years ago
  9. Mr.Math
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    You're welcome!

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

    • 2 years ago
  11. asnaseer
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    thanks @FoolForMath

    • 2 years ago
  12. FoolForMath
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    The complex number approach has been explained there too.

    • 2 years ago
  13. satellite73
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    i don't think there is a shorter way than series expansion

    • 2 years ago
  14. FoolForMath
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    @asnaseer: Glad to help :)

    • 2 years ago
  15. asnaseer
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    Using the uniqueness theorem with differentials seems to be the shortest method.

    • 2 years ago
  16. asnaseer
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    I guess it all depends on how you define "shortest"

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

    • 2 years ago
  18. asnaseer
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    I still find the proof using derivatives much simpler.

    • 2 years ago
  19. FoolForMath
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    Yes, I agree with you asnaseer.

    • 2 years ago
  20. asnaseer
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    it look more "elegant" as well.

    • 2 years ago
  21. asnaseer
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    *looks

    • 2 years ago
  22. FoolForMath
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    yea but inaccessible without knowledge of calculus.

    • 2 years ago
  23. JamesJ
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    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.

    • 2 years ago
  24. FoolForMath
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    btw asnaseer, what's your need?

    • 2 years ago
  25. asnaseer
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    no need really - I was just wondering if there were any other ways to prove this apart from the series expansion.

    • 2 years ago
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