1. Aug 5, 2011 #1

   inknit

   

   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
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3. Aug 6, 2011 #2

   Dr. Seafood

   

   I think essentially you've shown some intuition for why it's a true formula: both z = cos(x) + i·sin(x) and z = e^i·x solve the ODE dz/dx = i·z, and both are 1 when x = 0. By uniqueness of solutions to ODEs, we should have e^i·x = cos(x) + i·sin(x).

    

   Last edited: Aug 6, 2011

   Dr. Seafood, Aug 6, 2011
4. Aug 6, 2011 #3

   dalcde

   

   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

   

   Science Advisor

   Homework Helper

   hi inknit!
   

   nooo …
   
   ln(z) = ix + C …
   
   

   you still need to prove what C is! ​

    

   tiny-tim, Aug 6, 2011

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