Understanding formula components for Complex exponential sequence

Question

I am trying to understand how all components within the complex exponential sequence function fit together, I have been reading from the following slide

it is the line below 'then we can express' that confuses me. so far I have deduced the following (which may be incorrect):

My confusion lies in two areas:

1) The part of the function used for magnitude looks awfully similar to what I would expect to see for rotation in the complex plane, how has this been used for magnitude?

2) The part of the function used to denote the base, has been converted into another form of which I do not understand.

If anyone could shed any light on either of these point it would be very much appreciated. If it is able to be explained intuitively as opposed to algebraically this would be much better as I have found the visualisation of the components working together allow for a better overall understanding.

Answer 1

(1) They haven't used $e^{j\phi}$ to represent magnitude. They also didn't make any such claim elsewhere. You're the one somehow mistaking this for magnitude. This is a complex number, not a magnitude.

(2) I don't really understand what you mean here, but if you meant the expression for $\alpha,$ then it's not much different from the expression for $A.$ Here, we've simply expressed $\alpha$ in exponential form (this may be uniquely done for any nonzero complex number). That is, we write it in the form $e^{a+ib}=e^ae^{ib}=R(\cos b+i\sin b),$ where $R=|\alpha|$ is a positive real number. So this is still of the form $|\alpha|e^{i\psi}$ for suitable $\psi.$

Answer 2

The context of "sequence" suggest that $n$ should be an integer in $$ x[n]=A\alpha^n.\tag{0} $$

Your slide says if we write $$ \alpha=e^{\sigma_0+j\omega_0},\quad A=|A|e^{j\phi},\tag{1} $$ then $$ x[n]=A\alpha^n=(|A|e^{j\phi})\cdot (e^{\sigma_0+j\omega_0})^n =|A|e^{j\phi}e^{(\sigma_0+j\omega_0)n}.\tag{2} $$ This is nothing but substitution (1) into (0).

For your second question, check Euler's formula.