# Can we apply Euler's formula on plain waves?

I don't speak English so edit my question if it is not accurate.

Euler's formula for a complex number is: $$e^{i\theta}=\cos\theta + i \sin\theta$$ But when I write a plain wave as $$e^{i\vec{k}\cdot\vec{x}}=\cos(\vec{k}\cdot\vec{x}) + i \sin(\vec{k}\cdot\vec{x})$$ I think something is not right because $$\vec{k}\cdot\vec{x}=kx\cos(\beta)$$ that means we are taking the cosine of another cosine: Setting $$kx=1$$ we have $$e^{i\cos(\beta)}=\cos(\cos(\beta)) + i \sin(\cos(\beta))$$ which is a bit non-intuitive. So, I thought you may help me understanding $$e^{i\vec{k}\cdot\vec{x}}$$. Can we apply Euler's formula on plain waves? How did we extend an angle to a dot product?

• Ok, but $\beta$ isn't a variable, it is the angle between the x-axis and the wave propagation direction. – ProfRob Sep 20 '20 at 12:17

Setting $$kx=1$$ is probably what confuses you. If you go to a Cartesian frame you have $$\vec{k}\cdot \vec{x} = k_x x + k_y y + k_z z$$ which is a number. Let us call this number $$\ell$$. We then have $$e^{i\vec{k}\cdot \vec{x}}= e^{i\ell} = \cos \ell + i \sin \ell$$ So your formula is perfectly legitimate.
• Also worth mentioning: This $\ell$ is the phase difference between position $\vec{0}$ and position $\vec{x}$. – Thomas Fritsch Sep 20 '20 at 12:38