Noteworthy Uses:: *[[Noteworthy Uses]]* Proved By:: [[Euler's formula]] Intuition:: *[[Intuition]]* ----- > [!proposition] Proposition. ([[De Moivre's formula]]) > For any $x \in \rr$ and $n \in \zz$ we have $(\cos x + i\sin x)^{n} = \cos nx + i\sin nx.$ **Corollary**. We have $(re^{ix})^{n} = r^{n}\cis(nx) = r^{n}\cis^{n}x$. \ **Remark.** In fact, the proof given here shows that for all $n \in \cc$ the formula remains valid, but it is a little bit murky since when $n \notin \zz$ the expression $(e^{ix})^{n}$ is multivalued. In this case, we can write the formula in a way that makes such multivalued-ness explicit: for integer $k$ and $w \in \cc$, we have $(\cos x + i \sin x ) ^{w} = \cos \big(w(x+2 \pi k)\big) + i\sin\big(w(x + 2 \pi k)\big).$ (Notice why this is true but irrelevant if $w \in \zz$.) > [!proof]- Proof. ([[De Moivre's formula]]) > Using [[Euler's formula]], we have $\begin{align} (\cos x + i\sin x )^{n} = & (e^{ix})^{n} \\ = & e^{i(nx)} \\ = & \cos nx + i \sin nx. \end{align}$ as desired. The **corollary** is true because $(re^{ix})^{n}=r^{n}e^{i(nx)}=r^{n}\cis ^{n} x$ where the last equality uses [[De Moivre's formula]]. > [!intuition] > We can think of [[De Moivre's formula]] as a consequence of the fact that multiplication of [[complex numbers|complex numbers]] involves addition of their angles. For example, if we are just multiplying a number on the unit circle in the complex plane by itself $n-1$ times (i.e., raising it to the $n^{th}$ power), that's just adding up $n$ identical angles. ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```