# Quick modulus question - complex exponential function

#### buckylomax

##### New member
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.

#### ZaidAlyafey

##### Well-known member
MHB Math Helper
put z=a+ib then expand $$\displaystyle (a+ib)^n$$ using the binomial expansion ...

#### Klaas van Aarsen

##### MHB Seeker
Staff member
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.

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

#### buckylomax

##### New member
Welcome to MHB, buckylomax! That sounds fine to me.

$$\displaystyle |\exp(z^n)| = \exp(\Re(z^n)) \le \exp(|z^n|) = \exp(|z|^n) < \exp(1^n) = e$$
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 Thanks a lot for the help.

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