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Topic: analytic function problem
Replies: 76   Last Post: Oct 28, 1996 11:08 AM

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Zdislav V. Kovarik

Posts: 3,419
Registered: 12/6/04
Re: analytic function problem
Posted: Sep 30, 1996 1:02 PM
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In article <52o4t3$h95@ctylnk.cityu.edu.hk>,
Christopher <mawychan@sobolev.cityu.edu.hk> wrote:
>Hello,
>Please tell me how to prove simply that an analytic function is a function of
>z only, no z_bar.


First, you have to narrow it down to "continuously differentiable"
functions (and centuries ago, people didn't consider other functions
anyway).
Then you re-write the Cauchy-Riemann equations for w=u+iv (u and
v being real functions of real variables x, y):
(partial u)/(partial x) = (partial v)/(partial y)
(partial v)/(partial x) = - (partial u)/(partial y)
substituting formally x+iy = z
x-iy = z_bar
(inverse substitution: x=(z+z_bar)/2, y=(z-z_bar)/)2*i)

and presto! (partial w)/(partial z_bar) = 0

which, in the minds of the people I mentioned before, indicated that
"w is independent of z_bar".

There are gaps in the reasoning; to expose them, I recommend reading
"Counterexamples in Analysis" by B. Gelbaum and J. Olmsted, Holden-Day,
San Francisco 1964 (newer editions should exist), Chapter 9, Example 11.

Good luck, ZVK (Slavek).






Date Subject Author
9/30/96
Read analytic function problem
Christopher
9/30/96
Read Re: analytic function problem
David Ullrich
9/30/96
Read Re: analytic function problem
Hunter James D. STA x4202
10/1/96
Read Re: analytic function problem
David Ullrich
9/30/96
Read Re: analytic function problem
ilias kastanas 08-14-90
9/30/96
Read Re: analytic function problem
Dik T. Winter
10/1/96
Read Re: analytic function problem
David Kastrup
10/1/96
Read Re: analytic function problem
Hunter James D. STA x4202
9/30/96
Read Re: analytic function problem
Zdislav V. Kovarik
9/30/96
Read Re: analytic function problem
Anne DeCampo
10/2/96
Read Re: analytic function problem
Christopher
10/1/96
Read Re: analytic function problem
Tleko
10/2/96
Read Re: analytic function problem
David Kastrup
10/3/96
Read Re: analytic function problem
Dik T. Winter
10/3/96
Read Re: analytic function problem
Tleko
10/3/96
Read Re: analytic function problem
David Ullrich
10/3/96
Read Re: analytic function problem
Tleko
10/3/96
Read Re: analytic function problem
Dik T. Winter
10/3/96
Read Re: analytic function problem
Tleko
10/3/96
Read Re: analytic function problem
Zdislav V. Kovarik
10/5/96
Read Re: analytic function problem
Tleko
10/5/96
Read Re: analytic function problem
Dik T. Winter
10/6/96
Read Re: analytic function problem
Tleko
10/6/96
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David Ullrich
10/7/96
Read Re: analytic function problem
ilias kastanas 08-14-90
10/7/96
Read Re: analytic function problem
Tleko
10/7/96
Read Re: analytic function problem
Andreas Leitgeb
10/7/96
Read Re: analytic function problem
Andreas Leitgeb
10/7/96
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ilias kastanas 08-14-90
10/9/96
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Simon Read
10/8/96
Read Re: analytic function problem
Tleko
10/8/96
Read Re: analytic function problem
Zdislav V. Kovarik
10/8/96
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ilias kastanas 08-14-90
10/9/96
Read Re: analytic function problem
David Ullrich
10/8/96
Read Re: analytic function problem
Jim Hunter
10/8/96
Read Re: analytic function problem
Tleko
10/9/96
Read Re: analytic function problem
David Kastrup
10/9/96
Read Re: analytic function problem
Ilias Kastanas
10/11/96
Read Re: analytic function problem
Tleko
10/12/96
Read Re: analytic function problem
ilias kastanas 08-14-90
10/12/96
Read Re: analytic function problem
Dik T. Winter
10/11/96
Read Re: analytic function problem
Tleko
10/11/96
Read Re: analytic function problem
Tleko
10/11/96
Read Re: analytic function problem
Tleko
10/11/96
Read Re: analytic function problem
Tleko
10/11/96
Read Re: analytic function problem
Tleko
10/12/96
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Tleko
10/12/96
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Sue Franklin
10/13/96
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Ilias Kastanas
10/15/96
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Simon Read
10/13/96
Read Re: analytic function problem
Tleko
10/13/96
Read Re: analytic function problem
Tleko
10/13/96
Read Re: analytic function problem
ilias kastanas 08-14-90
10/14/96
Read Re: analytic function problem
Dik T. Winter
10/13/96
Read Re: analytic function problem
Tleko
10/13/96
Read Re: analytic function problem
Tleko
10/14/96
Read Re: analytic function problem
ilias kastanas 08-14-90
10/14/96
Read Re: analytic function problem
Dik T. Winter
10/15/96
Read Re: analytic function problem
Tleko
10/15/96
Read Re: analytic function problem
David Kastrup
10/15/96
Read Re: analytic function problem
Ilias Kastanas
10/16/96
Read Re: analytic function problem
Tleko
10/16/96
Read Re: analytic function problem
Dik T. Winter
10/17/96
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Raymond DeCampo
10/18/96
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Ariel Scolnicov
10/19/96
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Tleko
10/19/96
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Dik T. Winter
10/20/96
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Terry Moore
10/21/96
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Dik T. Winter
10/22/96
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Ariel Scolnicov
10/20/96
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Tleko
10/21/96
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Terry Moore
10/22/96
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David Kastrup
10/23/96
Read Re: analytic function problem
Tleko
10/24/96
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Gunter Bengel
10/28/96
Read Re: analytic function problem
David Kastrup
10/28/96
Read Re: analytic function problem
David Ullrich

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