MHB Quick modulus question - complex exponential function

[buckylomax]

What is |exp (z^n)| less than if |z| < 1? I'm thinking it's e but I'm having a brain freeze at the moment! Thanks for any help guys.

 

[alyafey22]

put z=a+ib then expand $$(a+ib)^n$$ using the binomial expansion ...

 

[I like Serena]

What is |exp (z^n)| less than if |z| < 1? I'm thinking it's e but I'm having a brain freeze at the moment! Thanks for any help guys.

Welcome to MHB, buckylomax! :)

That sounds fine to me.

$$|\exp(z^n)| = \exp(\Re(z^n)) \le \exp(|z^n|) = \exp(|z|^n) < \exp(1^n) = e$$

 

[buckylomax]

Welcome to MHB, buckylomax! :)

That sounds fine to me.

$$|\exp(z^n)| = \exp(\Re(z^n)) \le \exp(|z^n|) = \exp(|z|^n) < \exp(1^n) = e$$

That's what I was thinking but I just couldn't articulate it to the end. I think I broke my brain because I've been studying complex analysis for the past 8 hours (Puke)

Thanks a lot for the help.