Is this derivation of Euler's formula correct?

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Discussion Overview

The discussion centers around the derivation of Euler's formula, specifically the relationship between the complex exponential function and trigonometric functions. Participants explore the mathematical steps involved in the derivation and the implications of uniqueness in solutions to ordinary differential equations (ODEs).

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant presents a derivation showing that \( z = \cos(x) + i\sin(x) \) leads to \( e^{ix} = z \).
  • Another participant suggests that the derivation provides intuition for Euler's formula, noting that both forms solve the ODE \( \frac{dz}{dx} = iz \) and agree at \( x = 0 \).
  • Concerns are raised about the definition of the complex logarithm being crucial for the derivation's validity.
  • A later reply points out that the constant of integration \( C \) in the logarithmic expression needs to be established, indicating that the derivation is incomplete without this clarification.

Areas of Agreement / Disagreement

Participants express differing views on the completeness of the derivation, with some agreeing on the intuition behind Euler's formula while others highlight the need for a proper definition of the complex logarithm and the constant of integration.

Contextual Notes

There are unresolved issues regarding the definition of the complex logarithm and the determination of the constant of integration in the logarithmic expression.

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).
 
Last edited:
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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