Complex variables warmup

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

Complex variables warmup

Postby lime » Wed Aug 13, 2008 10:22 am

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
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Joined: Sun Jul 20, 2008 12:11 pm

Postby mathsubboy » Sun Aug 17, 2008 9:51 pm

I worked some drafts, is that the answer x^2=y^2? (where z=x+yi)

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lime
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Postby lime » Wed Aug 20, 2008 4:57 am

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.

Nirvana
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Joined: Sat Aug 23, 2008 3:57 pm

Postby Nirvana » Sat Aug 23, 2008 4:55 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

Postby mathsubboy » Wed Aug 27, 2008 9:58 am

I am sure the answer is wrong.

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

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lime
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Postby lime » Wed Aug 27, 2008 1:58 pm

Ok. What about this one.

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

mathsubboy
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Postby mathsubboy » Thu Aug 28, 2008 5:49 pm

0 is the only candidate satisfying CR equation.

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

so f is differentiable at zero.

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lime
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Postby lime » Thu Sep 11, 2008 1:57 am

Correct.




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