# Is this derivation of Euler's formula correct?

1. Aug 5, 2011

### 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)

2. Aug 6, 2011

### 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 = 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
3. Aug 6, 2011

### dalcde

Yes, but you have to make sure you have a good definition of the complex logarithm.

4. Aug 6, 2011

### tiny-tim

hi inknit!
nooo …

ln(z) = ix + C …

you still need to prove what C is!