Sketching regions in the complex plane and finding derivatives

[alexialight]

Just wondering if anyone could help me out with some problems I'm having with differential calculus.

Firstly, can anyone confirm if I sketched the region in image 1.jpg correctly (shown in 2.jpg)? I've done questions before where it just says |z|<2 and I know that it looks like a circle, but in this case the Re z at the end of the 2 just got me really confused.

Also, in image 3.jpg I have no idea as to how to even approach section a and I'm pretty sure I can't do b and c without understanding a. Can anyone help me out with this? Any hints or something because I've been staring at it for a while now and not getting anywhere (obviously).

 

[devious_]

For the first one, maybe writing z=x+iy (with Re(z)=x) will help.

For the second one, z=r(cost + isint) is called the modulus-argument form, i.e. r and t are the modulus and argument of z, respectively.

Does that help?

 

[dextercioby]

For the problem,make use of Euler's identity

$$e^{i\varphi}\equiv \cos\varphi+i\sin\varphi$$

and of the multivaluedness of the complex exponential.

Daniel.