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== Homework 2 collaboration area == | == Homework 2 collaboration area == | ||
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| − | <math>f(a)=\frac{1}{2\pi i}\int_\gamma \frac{f(z)}{z-a}\ dz.</math> | + | <math>f(a)=\frac{1}{2\pi i}\int_\gamma \frac{f(z)}{z-a}\ dz.</math> |
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| − | This is the place. | + | This is the place. |
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| − | + | '''Exercise 1''' | |
| − | <math> | + | Suppose that <math>\varphi(z)</math> is a continuous function on the trace of a path <span class="texhtml">γ</span>. Prove that the function |
| − | + | <math>f(z)=\int_{\gamma}\frac{\varphi(\zeta)}{\zeta-z}d\zeta</math> | |
| + | is analytic on <math>\mathbb{C}-\text{tr }\gamma</math>. | ||
| − | '''Discussion''' | + | <br> '''Discussion''' |
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| + | -A function is said to be analytic on an open set <span class="texhtml">Ω</span> if it is <math>\mathbb{C}</math>-differentiable at every point in <span class="texhtml">Ω</span>. Since the definition of analytic involves an open set, to complete Exercise 1 we need to have an open set somewhere. Since <span class="texhtml">tr γ</span> is the continuous image of a compact set, it is compact. Since <math>\mathbb{C}</math> is a Hausdorff space, <span class="texhtml">tr γ</span> is closed. Alternately one can note that <math>\mathbb{C}</math> is homeomorphic to <math>\mathbb{R}^{2}</math> where we know that a compact set is closed and bounded. Since the complement of a closed set is open, <math>\mathbb{C}-\text{tr }\gamma</math> is open. The long and short of this is: since<math>\mathbb{C}-\text{tr }\gamma</math> is an open set, one just has to show that <math>\varphi</math> is <math>\mathbb{C}</math>-differentiable at every point in this set in order to complete the exercise. | ||
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| − | [[ | + | |
| + | [[2014 Spring MA 530 Bell|Back to MA530, Spring 2014]] | ||
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| + | [[Category:MA5530Spring2014Bell]] [[Category:MA530]] [[Category:Math]] [[Category:Homework]] | ||
Revision as of 20:15, 28 January 2014
Homework 2 collaboration area
Here it is again:
$ f(a)=\frac{1}{2\pi i}\int_\gamma \frac{f(z)}{z-a}\ dz. $
This is the place.
Exercise 1
Suppose that $ \varphi(z) $ is a continuous function on the trace of a path γ. Prove that the function
$ f(z)=\int_{\gamma}\frac{\varphi(\zeta)}{\zeta-z}d\zeta $
is analytic on $ \mathbb{C}-\text{tr }\gamma $.
Discussion
-A function is said to be analytic on an open set Ω if it is $ \mathbb{C} $-differentiable at every point in Ω. Since the definition of analytic involves an open set, to complete Exercise 1 we need to have an open set somewhere. Since tr γ is the continuous image of a compact set, it is compact. Since $ \mathbb{C} $ is a Hausdorff space, tr γ is closed. Alternately one can note that $ \mathbb{C} $ is homeomorphic to $ \mathbb{R}^{2} $ where we know that a compact set is closed and bounded. Since the complement of a closed set is open, $ \mathbb{C}-\text{tr }\gamma $ is open. The long and short of this is: since$ \mathbb{C}-\text{tr }\gamma $ is an open set, one just has to show that $ \varphi $ is $ \mathbb{C} $-differentiable at every point in this set in order to complete the exercise.
