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What does it mean that $$w=\frac{az+b}{cz+d}$$ can map a circle to a line, a line to a line, a circle to a circle or a line to a circle?

Thanks for your explanation. I noticed many problems regarding Conformal Mapping are asking to form a map in order to get a desired result, for instance, fix some points or map a area to another one. But I don't know how can I find the desired mapping. Is there a certain rules I should follow? The following is an example. Let U be the upper half plane from which the points of the

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    is $a,b\in\mathbb{R}$ or $a,b\in\mathbb{C}$ ?2012-04-25
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    Here is a link to a very enlightening video. It may help build your intuition about what is behind these mappings. http://www.youtube.com/watch?v=JX3VmDgiFnY2012-04-25
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    @ John It truly is very enlightening an beautiful. Thank you.2012-04-25

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