Question

If \(z \in \mathbb C\), verify
\(\overline {e^z}= e^{\overline {z}}\)
Please, help

Answer 1

try using the series definition of e^z

Answer 2

\(e^z = \sum_{n=0}^\infty \dfrac{z^n}{n!}\)
right?

Answer 3

hint:
\[\Large \overline {{e^z}} = \overline {{e^{x + iy}}} = \overline {{e^x}{e^i}^y} = {e^x}{e^{ - i}}^y = ...?\]

Answer 4

oh, yeh!! because e^x is a real part, hence it doesn't change under the"bar" , right?

Answer 5

ha!! one line proof!!!

Answer 6

yes! furthermore we have the subsequent theorem:
\[\Large \overline {{z_1}{z_2}} = \overline {{z_1}} \;\overline {{z_2}} \]

Answer 7

Thanks so much.