# Derivation related to Euler’s Formula

#### mysearch

Gold Member
Hi,
I wasn’t sure whether to post this issue in a physics forum or here in this maths forum, because although it relates to physics is appears to be grounded in maths, i.e. Euler theorem. Therefore, I was wondering if anybody could help me resolve some issues with the following derivation, taken from a textbook, for 2 waves in superposition. The derivation is linked to the Young’s double-slit experiment, which acts as a precursor to quantum theory

 $$\Psi = A exp \; i \left[ k \left( r- \frac{d}{2}sin \theta\right) - wt \right] + A exp \; i \left[ k \left( r+ \frac{d}{2}sin \theta\right) - wt \right]$$

 $$\Psi = A exp \; i \left( kr-wt\right) \left( exp \left[ ik \left( \frac{d}{2} \right)sin \theta \right] + exp \left[ -ik \left( \frac{d}{2} \right) sin \theta \right] \right)$$

 $$\Psi = 2A exp \; i \left( kr-wt\right) cos \left[ k \left( \frac{d}{2} \right)sin \theta \right]$$

Purely, by way of background, equation  represents 2 waves in superposition, hence the 2 parts, separated by the distance between the 2 slits [d]. The distance [r] is the mean distance to the screen where the interference pattern is seen. However, my first maths question relates to the premise of equation  being based on the following relationship and whether equation  has to consider both the real and imaginary components of the wave function? (I accept this might not be a pure maths issue)

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

While I think I follow the basic steps from  to , there appears to be an assumption when going from  to  that:

 $$\left( exp \left[ ik \left( \frac{d}{2} \right)sin \theta \right] + exp \left[ -ik \left( \frac{d}{2} \right) sin \theta \right] \right) = 2 cos \left[ k \left( \frac{d}{2} \right)sin \theta \right]$$

So my second question is whether this based on the mathematical relationship?

 $$\Re (e^{i \theta}) = cos \theta = (e^{i \theta}+ e^{-i \theta})/2$$

In part, this goes back to the scope of whether equation  is intended to include the real and imaginary parts of the wave function. Would appreciate any knowledgeable insights to the maths and/or physics. Thanks

#### yyat

The formula

$$\cos(\theta)=\frac{e^{i\theta}+e^{-i\theta}}{2}$$

is true for all $$\theta\in\mathbb{C}$$, so  is equivalent to . Note though that

$$\Re{e^{i\theta}}=\cos(\theta)$$

is only true for real $$\theta$$.

#### mysearch

Gold Member
Note though that
$$\Re{e^{i\theta}}=\cos(\theta)$$
is only true for real $$\theta$$.
Thanks. I agree with your note, but wouldn’t this suggest that the whole of equation  has to be based on the real part, i.e. this equation is not considering the imaginary (+i.sin) component? I wasn't sure of the implication of this statement, but the issue may be more relevant to a physics forum addressing the wave function for mechanical and quantum waves.

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