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

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David Ullrich

Posts: 1,252
Registered: 12/6/04
Re: analytic function problem
Posted: Oct 1, 1996 10:45 AM
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Hunter James D. STA x4202 wrote:
>
> 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*".

>
> Yes, of course this is right.
>
> For any function of z, say, y = f(z) = f(conj(z*)).
>
> So, how about the following way to say this.
>
> If y = f(z) is an analytic function of z then y is necessarily
> not an analytic function of z*, and
> if y = g(z*) is an analytic function
> of z*, then y is necessarily not a function of z.
>
> In either case, I think you could use chain rule would show that dz*/dz exists,
> which can be easily shown to be false.
>
> If y is an analytic function of z then
> dy/dz = f'(z)
>
> and if y is an analytic function of z*
>
> dy/dz* = f'(z)dz/dz*
>
> Does this do it?


This occurred to me. It's not _quite_ correct, as Kastrup points out,
but it's easily correctable.
However I decided not to mention it because it seems circular, in a moral
if not formal sense: The question is why can an analytic function depend only
on z and not z*. This seems like we're sort of trying to determine what the
phrase "analytic function" really means, and if that's what we're trying to
do then an explanation that includes the phrase "analytic function" seems
a little off.

Seems like the "right" answer might have to do with power series:
Say you have a real-analytic function in the plane. This says it has a
power series expansion as a function in R^2, ie a power series expansion
in terms of x and y. It's easy to change that to a series in terms of
z and z*, and then it turns out the function is complex-analytic if and
only if there are no z*'s in the expansion. (The Cauchy-Riemann equations
say df/d(z*) = 0 (where the d's are supposed to be curly partial-derivative
things), and it turns out that if you have a power series in z and z* you
can calculate the "partial wrt z*" exactly as though z and z* were independent
variables and it was a real partial derivative...)

Hmm, I just realized I'm using different notation than you were here.
One often defines df/dz = (df/dx - i df/dy)/2 and df/d(z*) = (df/dx + i df/dy)/2.
Now these operators are defined for "any" f , and f is analytic if and only if
df/d(z*) = 0, in which case f' = df/dz . (Must not be exactly the notation
you had in mind, because, although I hesitate to mention this where people
might hear it, now d(z*)/dz certainly does exist, in fact d(z*)/dz = 0.
I hope nobody gets the idea I'm claiming z* is analytic...)

--
David Ullrich

?his ?s ?avid ?llrich's ?ig ?ile
(Someone undeleted it for me...)






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
Read Re: analytic function problem
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
Read Re: analytic function problem
ilias kastanas 08-14-90
10/9/96
Read Re: analytic function problem
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
Read Re: analytic function problem
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
Read Re: analytic function problem
Tleko
10/12/96
Read Re: analytic function problem
Sue Franklin
10/13/96
Read Re: analytic function problem
Ilias Kastanas
10/15/96
Read Re: analytic function problem
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
Read Re: analytic function problem
Raymond DeCampo
10/18/96
Read Re: analytic function problem
Ariel Scolnicov
10/19/96
Read Re: analytic function problem
Tleko
10/19/96
Read Re: analytic function problem
Dik T. Winter
10/20/96
Read Re: analytic function problem
Terry Moore
10/21/96
Read Re: analytic function problem
Dik T. Winter
10/22/96
Read Re: analytic function problem
Ariel Scolnicov
10/20/96
Read Re: analytic function problem
Tleko
10/21/96
Read Re: analytic function problem
Terry Moore
10/22/96
Read Re: analytic function problem
David Kastrup
10/23/96
Read Re: analytic function problem
Tleko
10/24/96
Read Re: analytic function problem
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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