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  1. What is the definition of angle in conformal mappings and how can we find angle between them for example if $\gamma_1(t)= \sin t$ and $\gamma_2(t)= \exp(it)$ then at $t=0$ find angle.
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please use LaTeX to typeset your formulas. Also, your $\gamma_1$ is just a function on the real line, your $\gamma_2$ a complex valued one (which one may interpret as a curve). In order to take about angles you need something like tangent vectors, e.g. tangents to differentible curves. – user20266 Jun 1 '12 at 14:51
I don't think your curves $\gamma_1$ and $\gamma_2$ intersect at $t = 0$. – froggie Jun 1 '12 at 15:06
I suspect something is missing in the definition of $\gamma_1$ -- as stated, this is a very odd way to parametrize the interval $[-1,1]$. – mrf Jun 1 '12 at 16:06
also I dont know more about this, but a very intelligent person asked me this question. also he asked me what is the definition of this angle, – Dr. V.S. Chauhan Jun 2 '12 at 12:45

Parametrise $z = z(t)$, with $z:(0,1) \to \mathbb{C}$. Then write $$w(z(t)) = f(z(t))$$ and consider by the chain rule $$\mathrm{arg}(\dot w(z(t))) = \mathrm{arg}(f'(z(t)) \dot z(t)) = \mathrm{arg}(f'(z(t))) + \mathrm{arg}( \dot z(t))$$

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I have no idea why you dug up a question that's over a year old, especially since it's so badly formulated. Look carefully at the OP's parametrization of $\gamma_1$. – mrf Jun 6 '13 at 22:12

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