Is this proof of Euler's Formula circular?

Question

I am looking at this proof (in the image) from a textbook for engineering mathematics:

I don't understand this proof, and my reasoning is as follows:

Known True Statements:

$z=r(cos\theta + isin\theta)$

$z_1z_2 = r_1r_2[cos(\theta_1 + \theta_2) + isin(\theta_1 + \theta_2)]$

$\frac{z_1}{z_2} = \frac{r_1}{r_2}[cos(\theta_1 - \theta_2) + isin(\theta_1 - \theta_2)]$

$e^{i\theta_1}e^{i\theta_2} = e^{i(\theta_1+\theta_2)}$

The line in the proof saying "When expressed in terms of Euler's formula, this becomes..." seems to me to be equivalent to:

IF $e^{i\theta} = cos\theta + isin\theta$, then

$e^{i\theta_1}e^{i\theta_1} = (cos\theta_1+isin\theta_1)(cos\theta_2+isin\theta_2)$

$e^{i\theta_1}e^{i\theta_1} = cos(\theta_1+\theta_2) + isin(\theta_1+\theta_2) = z_1z_2,$ where $r_1=r_2=1$

and:

$\frac{z_1}{z_2} = cos(\theta_1-\theta_2) + isin(\theta_1-\theta_2) = \frac{z_1}{z_2},$ where $r_1=r_2=1$

Therefore if Euler's formula is true, then it can be shown that $z = re^{i\theta}$, and since $z=r(cos\theta + isin\theta)$, it is finally shown that $e^{i\theta} = cos\theta + isin\theta$

I don't understand this proof because at the step when the proof says "When expressed in terms of Euler's formula this becomes...", I interpret this as meaning we assume the statement to be true. Am I correct in this assumption?

If that assumption is correct, is the proof circular because it assumes the statement is true in order to prove the statement?

Thanks very much!

Answer 1

This is not a proof of Euler's formula. Instead it is stating Euler's formula and using it to explain some otherwise difficult calculations that then become easier.

For example, we don't need to multiply out the real parts of two complex numbers, instead we can easily say $$ z_1z_2=r_1r_2e^{\theta_1 + \theta_2} $$

the same is done for dividing complex numbers in this example

Answer 2

If you put, formally, $e^{jx}=\cos x + j \sin x$ all the formulae work nicely (mathematicians use $i$ where physicists use $j$, by the way). That justifies using the expression.

However, this does not illuminate the particular value of $e$ - any number would give the same parallel between formulae considered formally. That is because the exponents add when expressions are multiplied and the formulae for the sine and cosine only depend on addition (or subtraction) of angles.

To identify $e$ with a particular constant you first need to give meaning to the expression $e^{jx}$, and then show that this meaning makes the equation work for Euler's constant $e$. There are various ways of doing this.

Note that the formula also depends for its beauty on the units used to measure the angle - radians must be used. There is a similar (but messier) formula for degrees and similarly for other angle measures.

The explanation given does not make any of this explicit, or prove it. What it does is show that the formula potentially makes sense.