Re: Complex Weirdness

From: Ramin (anonymous_at_mathforum.org)
Date: 09/08/04 

Date: Wed, 8 Sep 2004 15:20:51 +0000 (UTC)

On 08 Sep 2004, "Shmuel wrote:
>In <200409071919.i87JJR801378@proapp.mathforum.org>, on 09/07/2004
> at 07:25 PM, anonymous@mathforum.org (Ramin) said:
>
>> Hi. Could someone please help me explain why certain complex
>> expressions/functions such as x^2-y^2 +i2xy can be reduced to an
>>expression of z using the transformations Rez=z+z^-/2 and
>>Imz=z-z^-/2i (use z^- as conjugation), while others like xy+iy
>>cannot?.
>
>No, because it isn't true.
>
>> My understanding is that this means that x^2-y^2 +i2xy is
>> a function of a complex variable,( actually equal to z^2),
>> while xy+iy is not a function of a complex variable since it
>> is not reducible to an expression purely in terms of z.
>
>It is reducible to an expression purely in terms of z, and your first
>paragraph has the clue as to how to do that.

   Thanks for your reply.

  Could you please tell me how?. After trying to use the
 transforms above on xy+iy, I ended up with:

      (z^2-z^-2)/4i+(z-z^-) , and after manipulating it
     for a while, I still get nowhere.
    

>
>> My question, I guess, is, does this imply that there is
>> no map taking the complex plane to the set of points
>> (xy,y)?.
>
>No, the identity function is such a map. There is, in general, more
>than one map between a pair of sets.

   Do you mean the restriction of the identity map to this region?.
   I don't know if I am using some wrong definitions, but I thought
   I would get a function f with f asigning to z a point of the
    form xy+iy. f(z)=z , OTOH, maps z to x+iy

>
>> I know the Riemann mapping theorem gives the condition for
>> simply-connected regions;
>
>But not for arbitrary maps, or even arbitrary continuous maps. In
>general, when a theorem uses such terms as analytic, holomorphic or
>meromorphic, they are essential parts of that theorem; remove them and
>what is left will no longer be valid. The map _:x=iy->x-iy is not
>analytic.
>
>--
>Shmuel (Seymour J.) Metz, SysProg and JOAT <http://patriot.net/~shmuel>
>
>Unsolicited bulk E-mail subject to legal action. I reserve the
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