Upper bound for $|e^z|$

Question

I want to find an upper bound for $|e^z|$ on the circle $\gamma(t)=2e^{it}.$

My thoughts are as follows:

$$|e^z|=|e^{2(\cos t+i\sin t)}|=|e^{2\cos t}|\cdot|e^{2i\sin t}|\leq |e^2|\cdot|e^{2i}|=e^2$$

I'm not convinced that this is correct though. Is this right?

Maybe you should use Cauchy's integral formula for your cycle and use the estimation lemma to obtain the result. — Beslikas Thanos, Nov 26 '16 at 19:36

score 6 · Accepted Answer · answered Nov 26 '16 at 19:30

Yes, it's right. Note that in general you have $$|e^{z}| = |e^{\Re(z) + i\Im(z)}| = |e^{\Re(z)}|\cdot |e^{i\Im(z)}| = e^{\Re(z)}$$ where $\Re(z)$, $\Im(z)$ denote the real and imaginary parts of $z$. This gives $$|e^{z(t)}| = e^{2 \cos(t)} \leq e^{2}$$ in your example.

Upper bound for $|e^z|$

1 Answers1