Complex Analysis-Maximum Modulus of ze^z

[PhotonSSBM]

Homework Statement

Let

##f(z) = ze^z##

be bounded in the region where

## |z| \leq 2##, ##Im(z) \geq 0##, and ##Re(z) \geq 0##

Where does it achieve it's maximum modulus and what is that maximum modulus?

Homework Equations

N/A

The Attempt at a Solution

A theorem states that any function continuous in a bounded region up to and including the boundary achieves its maximum modulus on the boundary. f(z) has this property so we know it is on the boundary.

It obviously has to be somewhere on the circular curve, since |z| is 2 across the whole curve.

Another obvious thing is that it's max must be ##2e^2##

Now here is my question, do we treat where precisely it gains its max modulus in this way:
##|z|e^{|z|}##
or
##|z|e^x*e^{iy}##

because if we treat this like the first one, the whole circular curve is the max, whereas with the second one, only z=2 is the max.

Thanks for any help.

 

[BvU]

Why not look at ##f^*f = zz^* e^{z+z^*} ## ?

 

[PhotonSSBM]

##zz*e^{z+z*}=|z|e^{2Re(z)}##

And this implies that when Re(z) is maximum ##f* f## is also maximum since |z| = 2 everyone on the curve.

Gooooot it. Thankies.