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

 Messages: [ Previous | Next ]
 ilias kastanas 08-14-90 Posts: 443 Registered: 12/12/04
Re: analytic function problem
Posted: Sep 30, 1996 5:37 PM

In article <324FEC1A.4533@math.okstate.edu>,
David Ullrich <ullrich@math.okstate.edu> wrote:
@Christopher wrote:
@>
@> Hello,
@> Please tell me how to prove simply that an analytic function is a function of
@> z only, no z_bar.
@
@ Using z* for the conjugate of z:
@
@ You need to state the conclusion more precisely before this can be
@proved. We all agree that f(z) = z^2 is analytic, but f(z) = g(z*) for a
@certain function g , so f _is_ "a function of z*".

Let it be "you only need z, not z*" and try the following fun bit:

Say your function is f(x, y).
Since z = x+iy, z* = x-iy, we have x = (z + z*)/2, y = (z - z*)/2i.
Substitute into f, and compute the partial derivative: if df/dz* = 0
you are there!

E.g. f(x,y) = x^2 - y^2 + 2ixy = (z + z*)^2 /4 + (z - z*)^2 /4 +
+ (z^2 - (z*)^2)/2 and df/dz* = 0. Of course f simplifies to z^2.

Don't look for a limit definition of d/dz* ; this is for when you
can do it 'symbolically'. df/dz* = 0 is Cauchy-Riemann in disguise.

A similar approach yields from a harmonic u an f(z) with Re f = u.
Without integration or other old-fashioned stuff, what.

Ilias

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