Find a Mobius Transformation to Map Real Line to Unit Circle

[Amer]

Hey mobius transformation defined as
$$f(z) = \frac{az+b}{cz+d}$$
and $$ad \ne bc$$
it is a one to one function how i can find a mobius transformation that take the real line into the unit circle
I read it in the net
$$f(z) = \frac{z - i}{z+i}$$
and i checked it, it takes the real line into the unit circle, but there is a properties of the mobius transformation as the book said it is a combination of translation, inversion, rotation, dilation.

My question is how to find such map, or if we have the real line what first we have to do inversion,rotation,translation, ? to get the circle.

Thanks

 

[Ackbach]

Well, Mobius transformations take lines or circles to lines or circles. All you have to do is check three points.

$$f( \infty)=1, \quad f(0) = -1, \quad f(1)= \frac{1-i}{1+i}= \frac{1-i}{1+i} \cdot \frac{1-i}{1-i}
= \frac{1-2i-1}{1+1} = -i.$$

Therefore, you seem to have done it. This method you can use to do most any of these transformations. Take a line or circle into an appropriate line or circle by making sure your $a,b,c,d$ are chosen correctly. Then, if you must map a region, pick a point in the origin region, and make sure it winds up in the destination region.

 

[Amer]

still not clear, how did you determine $$f(\infty) = 1 , f(0) = -1$$
what I was thinking about I said
$$\mid f(0) \mid = 1 \\ \frac{\mid b \mid}{\mid d\mid } = 1 \\ \mid b \mid = \mid d\mid$$
thats one

then I found that $$\mid a \mid = \mid c \mid$$ by mapping infinity
after that guessing ?

what I was looking for is to master the inversion,translation, dilation, rotation
so I can imagine what i have to use to take a region to another

 

[Ackbach]

still not clear, how did you determine $$f(\infty) = 1 , f(0) = -1$$

Technically, the $f( \infty)$ is the limit:
$$ \lim_{x \to \infty}f(z)= \lim_{z \to \infty} \frac{z-i}{z+i}
= \lim_{z \to \infty} \frac{1-i/z}{1+i/z}=1.$$

I determined to check $0, 1, \infty$, because those are easy values to check on the real line.

what I was thinking about I said
$$\mid f(0) \mid = 1 \\ \frac{\mid b \mid}{\mid d\mid } = 1 \\ \mid b \mid = \mid d\mid$$
thats one

then I found that $$\mid a \mid = \mid c \mid$$ by mapping infinity
after that guessing ?

what I was looking for is to master the inversion,translation, dilation, rotation
so I can imagine what i have to use to take a region to another

I've always just transformed the boundaries of regions, and made sure the inside of the region gets mapped correctly. You can check out inversions, translations, etc., here.

 

[Amer]

Thanks very :D