I was messing around and decided to prove Euler's 'formula' using a method that doesn't involve power series. Here's how I did it:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]z=\cos\theta + i \sin\theta[/tex]

[tex]\frac{dz}{d\theta} = -\sin\theta + i\cos\theta[/tex]

[tex]i\frac{dz}{d\theta} = -i\sin\theta + i^{2}\cos\theta = -i\sin\theta - \cos\theta = -(\cos\theta+i\sin\theta) = -z[/tex]

[tex]-i\int \frac{1}{z} \; dz = \int \; d\theta[/tex]

[tex]-i \log|z| = \theta + C[/tex]

When [itex]\theta=0[/tex], [itex]z=1[/itex], so [itex]C=0[/itex]. Now:

[tex]\log|z| = \frac{-\theta}{i}[/tex]

[tex]z=e^{\frac{-\theta}{i}}[/tex]

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# Where'd I go wrong?

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