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 \(\displaystyle (a+ib)^n\) 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.

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

 

[buckylomax]

Welcome to MHB, buckylomax! :)

That sounds fine to me.

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

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.

 

[blue_raver22]

Great question! The quick modulus question you have posed involves the complex exponential function, which is given by exp(z) = e^z. The modulus, or absolute value, of a complex number z is denoted by |z| and is equal to the distance of z from the origin on the complex plane. In this case, we are looking for the modulus of the complex exponential function exp(z^n) where n is any integer.

Now, let's consider the condition |z| < 1. This means that the absolute value of z is less than 1, which can also be interpreted as z lying within the unit circle on the complex plane. In other words, z is closer to the origin than it is to any other point on the complex plane.

With this in mind, we can see that for any complex number z lying within the unit circle, the value of exp(z) will also lie within the unit circle. This is because as z gets closer to the origin, e^z gets closer to 1, and as z gets further away from the origin, e^z gets larger and larger. Therefore, the modulus of exp(z) will always be less than or equal to 1.

Now, let's apply this concept to the complex exponential function exp(z^n). Since n is an integer, we can think of this function as repeatedly multiplying exp(z) by itself n times. Therefore, the value of exp(z^n) will also lie within the unit circle for any complex number z lying within the unit circle. This means that the modulus of exp(z^n) will always be less than or equal to 1.

In conclusion, the answer to your question is that |exp(z^n)| is always less than or equal to 1 if |z| < 1. This is because the complex exponential function exp(z^n) maps any complex number z within the unit circle onto a point within the unit circle, resulting in a modulus that is always less than or equal to 1. I hope this helps to clarify your understanding. Keep up the great thinking!