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 Post subject: Complex Analysis - Question about methodology for a mapping.Posted: Tue, 13 Mar 2012 08:47:46 UTC

Joined: Thu, 1 Mar 2012 10:14:08 UTC
Posts: 7
Show that the image of the open disk |z + 1 + i | < 1 under the transformation w = (3 - 4i)z + 6 + 2i is the open disk |w + 1 - 3i| < 5.

The method done by the book is as follows, through an inverse transformation:

z = (w - 6 - 2i)/(3 - 4i)

This is substituted into the first given inequality.

Simplifying down to |w - 6 - 2i + (1 + i)(3 - 4i)| < 5, which now goes to |w + 1 - 3i| < 5.

My method was to use the transformation w = .... directly and substitute that into |w + 1 - 3i| < 5

Doing it this way and simplifying it, we're left with the initial inequality |z + 1 + i| < 1, which means it's correct, but is the method right?

I do not fully understand this inverse transformation idea. If we have something (a point/a plane/whatever) in the z-plane, and want to show the corresponding image of it in the w-plane, why are we using an inverse transformation z = ... ?

I cannot get my head around this. Shouldn't we just the actual mapping function w = ... to prove that the image is the correct one?

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 Post subject: Re: Complex Analysis - Question about methodology for a mappPosted: Tue, 13 Mar 2012 15:25:07 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6007
Location: 127.0.0.1, ::1 (avatar courtesy of UDN)
Cryptid wrote:
Show that the image of the open disk |z + 1 + i | < 1 under the transformation w = (3 - 4i)z + 6 + 2i is the open disk |w + 1 - 3i| < 5.

The method done by the book is as follows, through an inverse transformation:

z = (w - 6 - 2i)/(3 - 4i)

This is substituted into the first given inequality.

Simplifying down to |w - 6 - 2i + (1 + i)(3 - 4i)| < 5, which now goes to |w + 1 - 3i| < 5.

My method was to use the transformation w = .... directly and substitute that into |w + 1 - 3i| < 5

Doing it this way and simplifying it, we're left with the initial inequality |z + 1 + i| < 1, which means it's correct, but is the method right?

I do not fully understand this inverse transformation idea. If we have something (a point/a plane/whatever) in the z-plane, and want to show the corresponding image of it in the w-plane, why are we using an inverse transformation z = ... ?

I cannot get my head around this. Shouldn't we just the actual mapping function w = ... to prove that the image is the correct one?

Unfortunately not, seeing that you did not mention the magic word.

_________________

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 Post subject: Re: Complex Analysis - Question about methodology for a mappPosted: Tue, 13 Mar 2012 20:05:19 UTC

Joined: Thu, 1 Mar 2012 10:14:08 UTC
Posts: 7
Sorry, what word is that?

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 Post subject: Re: Complex Analysis - Question about methodology for a mappPosted: Sat, 17 Mar 2012 08:33:50 UTC

Joined: Thu, 1 Mar 2012 10:14:08 UTC
Posts: 7
Anyone have an idea regarding this?

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