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Complex variables warmup

Posted: Wed Aug 13, 2008 10:22 am
by lime
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)

Posted: Sun Aug 17, 2008 9:51 pm
by mathsubboy
I worked some drafts, is that the answer x^2=y^2? (where z=x+yi)

Posted: Wed Aug 20, 2008 4:57 am
by lime
Here is my solution.
Image
which implies that f(z) is differentiable only in point z=0.
The answer though is that f(z) is not differentiable everywhere. :D Would appreciate if someone pointed the mistake in my solution.

Posted: Sat Aug 23, 2008 4:55 pm
by Nirvana
I think your solution is right, lime. Probably some mistakes in an answer...

yes

Posted: Wed Aug 27, 2008 9:58 am
by mathsubboy
I am sure the answer is wrong.

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

Posted: Wed Aug 27, 2008 1:58 pm
by lime
Ok. What about this one.

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

Posted: Thu Aug 28, 2008 5:49 pm
by mathsubboy
0 is the only candidate satisfying CR equation.

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

so f is differentiable at zero.

Posted: Thu Sep 11, 2008 1:57 am
by lime
Correct.