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Post by Admin on Aug 12, 2016 6:30:18 GMT
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Gary
GaryVasco
Posts: 3,352 
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Post by Gary on Aug 17, 2016 23:48:41 GMT
Vasco,
I find no fault with Ch 6, Ex 5.
We have shown that $\frac{dR^{\psi}_{a}(z)}{dz} = \frac{|a|^2 + |b|^2}{(-\overline{B}z + \overline{A})^2}$ and that $|f'(z)| = \frac{|a|^2 + |b|^2}{|-\overline{B}z + \overline{A}|^2}$, so it appears that we have shown that the Mobius transformation (20) satisfies the differential equation $|f'(z)| = \frac{|a|^2 + |b|^2}{|-\overline{B}z + \overline{A}|^2}$ rather than (17). Does it follow that it also satisfies (17) as printed? Perhaps not. Since you qualified your answer in the original message, I assume you agree.
Gary
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Post by Admin on Aug 18, 2016 6:40:59 GMT
Vasco, I find no fault with Ch 6, Ex 5. We have shown that $\frac{dR^{\psi}_{a}(z)}{dz} = \frac{|a|^2 + |b|^2}{(-\overline{B}z + \overline{A})^2}$ and that $|f'(z)| = \frac{|a|^2 + |b|^2}{|-\overline{B}z + \overline{A}|^2}$, so it appears that we have shown that the Mobius transformation (20) satisfies the differential equation $|f'(z)| = \frac{|a|^2 + |b|^2}{|-\overline{B}z + \overline{A}|^2}$ rather than (20)(17). Does it follow that it also satisfies (20)(17) as printed? Perhaps not. Since you qualified your answer in the original message, I assume you agree. Gary Gary After thinking about your post and composing my reply and publishing it, I suddenly realised that I had very probably completely misconstrued your post, and so this is a very different post than the original, which you probably didn't read anyway. Your first sentence above made me think that you disagreed with me about the mistake in (17) despite your earlier post agreeing with me, and this coloured all my subsequent comments about this post. In what follows I am assuming that this comment refers to my published solution and not to the statement of the exercise!!. In my solution I also show that $\frac{|A|^2 + |B|^2}{|-\overline{B}z + \overline{A}|^2}=\frac{1+|f|^2}{1+|z|^2}$ which leads us to (17), if we apply my suggested correction to the statement of (17). Shouldn't your last two references to (20), in your post, be references to (17)? No, because if we know only the modulus (length) of a complex number $w$, we cannot deduce what the complex number $w$ is. Vasco |
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Gary
GaryVasco
Posts: 3,352
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Post by Gary on Aug 18, 2016 16:59:03 GMT
Vasco,
Sorry about the confusion of (17) and (20). I repaired the original.
As for the gist of it, I think we are in complete agreement.
Gary
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Post by Admin on Aug 18, 2016 17:33:02 GMT
Gary
Thanks, no problem.
Vasco
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