Is this derivation of Euler's formula correct?

AI Thread Summary
The discussion centers on the derivation of Euler's formula, expressed as z = cos(x) + isin(x) and its equivalence to e^(ix). Participants agree that both forms solve the ordinary differential equation dz/dx = i·z and are equal at x = 0, suggesting a unique solution. However, there is a debate regarding the definition of the complex logarithm and the constant of integration, C, in the equation ln(z) = ix + C. Clarification on the value of C is deemed necessary for a complete proof. The conversation emphasizes the importance of rigorous definitions in complex analysis.
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)
 
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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).
 
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Yes, but you have to make sure you have a good definition of the complex logarithm.
 
hi inknit! :smile:
inknit said:
∫ dz/z = ∫ idx

ln(z) = ix

nooo …

ln(z) = ix + C …

you still need to prove what C is! :wink:​
 

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