| Version 7 |
Version 6 |
| Euler's relation (also known as Euler's formula) is considered the first bridge between the fields of algebra and geometry, as it relates the exponential function to the trigonometric sine and cosine functions. |
Euler's relation (also known as Euler's formula) is considered the first bridge between the fields of algebra and geometry, as it relates the exponential function to the trigonometric sine and cosine functions. |
| The goal is to prove |
The goal is to prove |
| e^{ix} = \cos (x) + i\sin (x) |
e^{ix} = \cos (x) + i\sin (x) |
| It's easy to show that |
It's easy to show that |
| \begin{eqnarray*} |
\begin{eqnarray*} |
| i^{4n} & = & 1 \\ |
i^{4n} & = & 1 \\ |
| i^{4n+1} & = & i \\ |
i^{4n+1} & = & i \\ |
| i^{4n+2} & = & -1 \\ |
i^{4n+2} & = & -1 \\ |
| i^{4n+3} & = & -i |
i^{4n+3} & = & -i |
| \end{eqnarray*} |
\end{eqnarray*} |
| Now, using the Taylor series expansions of $\sin x$, $\cos x$ and $e^x$, we can show that |
Now, using the Taylor series expansions of $\sin x$, $\cos x$ and $e^x$, we can show that |
| \begin{eqnarray*} |
\begin{eqnarray*} |
| e^{ix} & = & \sum_{n=0}^{\infty} \frac{i^n x^n}{n!} \\ |
e^{ix} & = & \sum_{n=0}^{\infty} \frac{i^n x^n}{n!} \\ |
| e^{ix} & = & \sum_{n=0}^{\infty} \frac{x^{4n}}{(4n)!} + \frac{ix^{4n+1}}{(4n+1)!} |
e^{ix} & = & \sum_{n=0}^{\infty} \frac{x^{4n}}{(4n)!} + \frac{ix^{4n+1}}{(4n+1)!} |
| -\frac{x^{4n+2}}{(4n+2)!} - \frac{ix^{4n+3}}{(4n+3)!} |
-\frac{x^{4n+2}}{(4n+2)!} - \frac{ix^{4n+3}}{(4n+3)!} |
| \end{eqnarray*} |
\end{eqnarray*} |
| Because the series expansion above is absolutely convergent for all $x$, |
Because the series expansion above is absolutely convergent for all $x$, |
| we can rearrange the terms of the series as follows |
we can rearrange the terms of the series as follows |
| \begin{eqnarray*} |
\begin{eqnarray*} |
| e^{ix} & = & \sum_{n=0}^{\infty} \frac{x^{2n}}{(2n)!}(-1)^n + |
e^{ix} & = & \sum_{n=0}^{\infty} \frac{x^{2n}}{(2n)!}(-1)^n + |
| i \sum_{n=0}^{\infty} \frac{x^{2n+1}}{(2n+1)!}(-1)^n\\ |
i \sum_{n=0}^{\infty} \frac{x^{2n+1}}{(2n+1)!}(-1)^n\\ |
| e^{ix} & = & \cos(x) + i\sin(x) |
e^{ix} & = & \cos(x) + i\sin(x) |
| \end{eqnarray*} |
\end{eqnarray*} |
