help for two questions please

mami · **Joined:** Tue, 29 Nov 2011 19:41:49 UTC **Posts:** 13

I couldn't tackle these two problems. if you don't mind, could you help for these exercises... thanks for everything.

1-Denfie the oriented angle between two curves passing through a point in the complex plane. Show that an analytic function with non zero derivative preserves oriented angles.

2-Let $\Omega$ be a domain in the complex plane and suppose that f is an analytic function on $\Omega$ . Set $\Omega ^{*}=(z:\bar{z}\Omega)$ Show that the function g denfied by $g(z)=\bar{f(\bar{z})}$ is analytic in the domain $\Omega ^{*}$

Shadow · **Posted:** Sun, 19 Feb 2012 22:10:42 UTC

mami wrote:

I couldn't tackle these two problems. if you don't mind, could you help for these exercises... thanks for everything.

1-Denfie the oriented angle between two curves passing through a point in the complex plane. Show that an analytic function with non zero derivative preserves oriented angles.

2-Let $\Omega$ be a domain in the complex plane and suppose that f is an analytic function on $\Omega$ . Set $\Omega ^{*}=(z:\bar{z}\Omega)$ Show that the function g denfied by $g(z)=\bar{f(\bar{z})}$ is analytic in the domain $\Omega ^{*}$

You should be able to find the proof of 1 in any good complex analysis book at all.

For the second one, I think you mean $\Omega^*=\{z\in\mathbb{C}\; |\; \overline{z}\in\Omega\}$ can you confirm this?

If that's the case then, it's easy, just look at the power series and use the Cauchy-Hadamard theorem plus the fact that

is analytic on $\Omega$ .

mami · **Joined:** Tue, 29 Nov 2011 19:41:49 UTC **Posts:** 13

Yes. You rigth. I mean $\Omega^*=\{z\in\mathbb{C}\; |\; \overline{z}\in\Omega\}$
And i have a little request. Could you propose a book which i can find the proof of first question. Thanks.

Shadow wrote:

mami wrote:

I couldn't tackle these two problems. if you don't mind, could you help for these exercises... thanks for everything.

1-Denfie the oriented angle between two curves passing through a point in the complex plane. Show that an analytic function with non zero derivative preserves oriented angles.

2-Let $\Omega$ be a domain in the complex plane and suppose that f is an analytic function on $\Omega$ . Set $\Omega ^{*}=(z:\bar{z}\Omega)$ Show that the function g denfied by $g(z)=\bar{f(\bar{z})}$ is analytic in the domain $\Omega ^{*}$

You should be able to find the proof of 1 in any good complex analysis book at all.

For the second one, I think you mean $\Omega^*=\{z\in\mathbb{C}\; |\; \overline{z}\in\Omega\}$ can you confirm this?

If that's the case then, it's easy, just look at the power series and use the Cauchy-Hadamard theorem plus the fact that

is analytic on $\Omega$ .

Shadow · **Posted:** Mon, 20 Feb 2012 21:01:20 UTC

mami wrote:

Yes. You rigth. I mean $\Omega^*=\{z\in\mathbb{C}\; |\; \overline{z}\in\Omega\}$
And i have a little request. Could you propose a book which i can find the proof of first question. Thanks.

Shadow wrote:

mami wrote:

I couldn't tackle these two problems. if you don't mind, could you help for these exercises... thanks for everything.

1-Denfie the oriented angle between two curves passing through a point in the complex plane. Show that an analytic function with non zero derivative preserves oriented angles.

2-Let $\Omega$ be a domain in the complex plane and suppose that f is an analytic function on $\Omega$ . Set $\Omega ^{*}=(z:\bar{z}\Omega)$ Show that the function g denfied by $g(z)=\bar{f(\bar{z})}$ is analytic in the domain $\Omega ^{*}$

You should be able to find the proof of 1 in any good complex analysis book at all.

For the second one, I think you mean $\Omega^*=\{z\in\mathbb{C}\; |\; \overline{z}\in\Omega\}$ can you confirm this?

If that's the case then, it's easy, just look at the power series and use the Cauchy-Hadamard theorem plus the fact that

is analytic on $\Omega$ .

Ahlfors has it.

S.O.S. Mathematics CyberBoard

help for two questions please

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