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 - The Fusion of Science and Community**

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

# Where'd I go wrong?

Loading...

Similar Threads - Where'd wrong | Date |
---|---|

I Fourier Series: I don't understand where I am wrong -- please help | Jun 16, 2016 |

Integration where am I going wrong? | Jan 15, 2016 |

Tangent to a a curve, something seems wrong (Calculus) | Jun 25, 2015 |

Weird integral - what's wrong? | May 17, 2015 |

**Physics Forums - The Fusion of Science and Community**