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Quick modulus question - complex exponential function

buckylomax

New member
Aug 3, 2012
3
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
Jan 17, 2013
1,667
put z=a+ib then expand \(\displaystyle (a+ib)^n\) using the binomial expansion ...
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,780
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
Aug 3, 2012
3
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 (Puke)

Thanks a lot for the help.
 
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