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Is this derivation of Euler's formula correct?

  1. Aug 5, 2011 #1

    inknit

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    z = cos(x) + isin(x)
    dz = -sin(x) + icos(x)dx
    = i(isin(x) + cos(x))dx

    ∫ dz/z = ∫ idx

    ln(z) = ix

    e^(ix) = z

    e^(ix) = cos(x) + isin(x)
     
    inknit, Aug 5, 2011
  2. jcsd
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  3. Aug 6, 2011 #2

    Dr. Seafood

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    I think essentially you've shown some intuition for why it's a true formula: both z = cos(x) + i·sin(x) and z = ei·x solve the ODE dz/dx = i·z, and both are 1 when x = 0. By uniqueness of solutions to ODEs, we should have ei·x = cos(x) + i·sin(x).
     
    Last edited: Aug 6, 2011
    Dr. Seafood, Aug 6, 2011
  4. Aug 6, 2011 #3

    dalcde

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    Yes, but you have to make sure you have a good definition of the complex logarithm.
     
    dalcde, Aug 6, 2011
  5. Aug 6, 2011 #4

    tiny-tim

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    hi inknit! :smile:
    nooo …

    ln(z) = ix + C …

    you still need to prove what C is! :wink:​
     
    tiny-tim, Aug 6, 2011
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