## Complex variables warmup

Forum for the GRE subject test in mathematics.
lime
Posts: 129
Joined: Tue Dec 04, 2007 2:11 am

### Complex variables warmup

Since we started very good warmup in Abstract algebra I think it would be also useful to create the same topic for Complex variables. If you have any interesting problems in this field please do not hesitate and post it.

Problem 1. Find the points where the derivative of function
f(z) = (z^2)*(z_)
exist. (here "z_" means conjugate of z)

mathsubboy
Posts: 11
Joined: Sun Jul 20, 2008 12:11 pm
I worked some drafts, is that the answer x^2=y^2? (where z=x+yi)

lime
Posts: 129
Joined: Tue Dec 04, 2007 2:11 am
Here is my solution.

which implies that f(z) is differentiable only in point z=0.
The answer though is that f(z) is not differentiable everywhere. Would appreciate if someone pointed the mistake in my solution.

Nirvana
Posts: 1
Joined: Sat Aug 23, 2008 3:57 pm
I think your solution is right, lime. Probably some mistakes in an answer...

mathsubboy
Posts: 11
Joined: Sun Jul 20, 2008 12:11 pm

### yes

I am sure the answer is wrong.

I checked the all conditions, the function is definitely analytic on the
complex plane------including zero.

lime
Posts: 129
Joined: Tue Dec 04, 2007 2:11 am

Problem 2. f(z)=|z|^2

mathsubboy
Posts: 11
Joined: Sun Jul 20, 2008 12:11 pm
0 is the only candidate satisfying CR equation.

and |z|^2=z*(z_)

so f is differentiable at zero.

lime
Posts: 129
Joined: Tue Dec 04, 2007 2:11 am
Correct.