Euler's Formula/Proof

Theorem

 * $e^{i \theta} = \cos \theta + i \sin \theta$

where $e^\cdot$ is the complex exponential function, $\cos$ is cosine, $\sin$ is sine, and $i$ is the imaginary unit.

Thus we define the complex exponential function in terms of standard trigonometric functions.

Direct Proof 3
Use the Taylor Series Expansion for Exponential Function, we have:


 * $\displaystyle e^{i \theta} = 1 + i \theta + \frac {i^2\theta^2} {2!} + \frac {i^3 \theta^3} {3!} + \frac {i^4 \theta^4} {4!} + \frac {i^5\theta^5} {5!} + \frac {i^6 \theta^6} {6!} + \frac {i^7 \theta^7} {7!} + \frac {i^8 \theta^8} {8!} + \cdots$

The equation can be simplified to


 * $\displaystyle e^{i \theta} = 1 + i \theta - \frac {\theta^2} {2!} - \frac {i \theta^3} {3!} + \frac {\theta^4} {4!} + \frac {i \theta^5} {5!} - \frac {\theta^6} {6!} - \frac {i \theta^7}{7!} + \frac{\theta^8}{8!} + \cdots$

Rearranging the above equation, we obtain

We recognize the terms from the definitions of the sine and cosine functions:

As Sine Function is Absolutely Convergent and Cosine Function is Absolutely Convergent, it follows that the the rearranging of the Taylor Series Expansion for Exponential Function was allowed.

Hence:
 * $ e^{i \theta} = \cos \theta + i \sin \theta$