- #1
futurebird
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I have some questions about this proof. I have numbered the equations (1), (2), ... so I can ask about them.
THEOREM
If the piecewise differentiable closed curve [tex]\gamma[/tex] does not pass through the point a then:
[tex](1) \displaystyle\oint_{\gamma}\frac{dz}{z-a}[/tex]
is s multiple of [tex]2 \pi i[/tex]
PROOF
[tex]\gamma[/tex] is given by z(t) [tex]\alpha \leq t \leq \beta[/tex] consider:
[tex](2) h(t)=\displaystyle\int^{t}_{\aplha}\frac{z'(t)}{z(t)-a}dt[/tex]
h(t) is defined and continuos on the closed interval [tex](\alpha, \beta)[/tex]
[tex](3) h'(t)=\frac{z'(t)}{z(t)-a}[/tex]
where z'(t) is continuos.
[tex](4) k=e^{-h(t)}(z(t)-a)[/tex]
(5) Hence k' = 0.
[tex](6) e^{h(t)}=\frac{(z(t)-a)}{k}[/tex]
[tex](7) e^{h(t)}=\frac{(z(t)-a)}{z(\alpha)-a}[/tex]
Since [tex]\gamma[/tex] is a closed curve [tex]z(\beta)=z(\alpha)[/tex]
[tex](8) e^{h(\beta)}=1[/tex]
[tex]h(\beta)[/tex] must be a multiple of [tex]2 \pi i[/tex].
END
QUESTIONS
1. (3) why do we need to know about h'(t)?
2. (5) why is k' = 0?
3. How did we replace k with [tex]z(\alpha)-a[/tex] in step (7) ?
Thanks for any help you can give. I need to understand this proof for a test and I'd rather not memorize these bits, but instead know what I'm doing! Thanks!
THEOREM
If the piecewise differentiable closed curve [tex]\gamma[/tex] does not pass through the point a then:
[tex](1) \displaystyle\oint_{\gamma}\frac{dz}{z-a}[/tex]
is s multiple of [tex]2 \pi i[/tex]
PROOF
[tex]\gamma[/tex] is given by z(t) [tex]\alpha \leq t \leq \beta[/tex] consider:
[tex](2) h(t)=\displaystyle\int^{t}_{\aplha}\frac{z'(t)}{z(t)-a}dt[/tex]
h(t) is defined and continuos on the closed interval [tex](\alpha, \beta)[/tex]
[tex](3) h'(t)=\frac{z'(t)}{z(t)-a}[/tex]
where z'(t) is continuos.
[tex](4) k=e^{-h(t)}(z(t)-a)[/tex]
(5) Hence k' = 0.
[tex](6) e^{h(t)}=\frac{(z(t)-a)}{k}[/tex]
[tex](7) e^{h(t)}=\frac{(z(t)-a)}{z(\alpha)-a}[/tex]
Since [tex]\gamma[/tex] is a closed curve [tex]z(\beta)=z(\alpha)[/tex]
[tex](8) e^{h(\beta)}=1[/tex]
[tex]h(\beta)[/tex] must be a multiple of [tex]2 \pi i[/tex].
END
QUESTIONS
1. (3) why do we need to know about h'(t)?
2. (5) why is k' = 0?
3. How did we replace k with [tex]z(\alpha)-a[/tex] in step (7) ?
Thanks for any help you can give. I need to understand this proof for a test and I'd rather not memorize these bits, but instead know what I'm doing! Thanks!