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]

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Where'd I go wrong?

**Physics Forums | Science Articles, Homework Help, Discussion**