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 | | From: | emailzul at starhub.net.sg | | Subject: | Complex theorems on Limits | | Date: | 23 Jan 2005 07:36:17 -0800 |
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 | A theorem on limits (complex) states that
f(Z)= u(x,y) + iv(x,y), Zo = Xo + iYo, Wo=Uo+iVo
Then
lim f(Z)= Wo as Z tends to Zo
if and only if
lim u(x,y) = Uo as (x,y) tends to (Xo,Yo) and
lim v(x,y) = Vo as (x,y) tends to (Xo,Yo)
Therefore I need to show,
For any eps>0, there exist delta>0 such that
|u(x,y) - Uo| < eps whenever 0<|Z-Zo|< eps
|v(x,y) - Vo| < eps whenever 0<|Z-Zo|< eps
Proof:
let eps>0 since f(Z)=Wo as Z tends to Zo , there exist a delta>0 such
that,
|f(z) - Wo|< eps
|u(x,y) + iv(x,y) - Uo-iVo| < eps
|(u(x,y) - Uo) + i(v(x,y)-Vo)| < eps
| Re {u(x,y) - Uo + i(v(x,y) - Vo)}| < eps
Therefore
| u(x,y) - Uo| < eps likewise |v(x,y) - Vo| < eps whenever
0<|Z-Zo|< eps
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After this is where I need help, I will appreciate it if anyone could
show me the rest of the step to show that the converse of the theorem
is true due to the iff statement.
Regards
Zul
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| | From: | José_Carlos_Santos | | Subject: | Re: Complex theorems on Limits | | Date: | Sun, 23 Jan 2005 15:50:12 +0000 |
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| emailzul@starhub.net.sg wrote:
> After this is where I need help, I will appreciate it if anyone could
> show me the rest of the step to show that the converse of the theorem
> is true due to the iff statement.
If a and b are real numbers and if |a|,|b| < eps/sqrt(2), then
|a + bi| < eps.
I hope that this helps.
Best regards,
Jose Carlos Santos
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