Conformal map from the region outside a semicircle to the region outside a disk

[Supercazzola]

Show how to get a conformal map from the region outside a semicircle,

C∖SR={(x1,x2):|x1|2+|x2|2=R2,x2≤0}

to the region outside a disk D of radius

R2–√

centered at the origin,

C∖D

, ending up with

h(u)=iR+u+i⋅R2−u2−−−−−−−√2

, with

u=x1+ix2

.

I know that the idea is to use a Moebius transform to send the semicircle to two perpendicular lines through the origin (something like

az+Rz−R

) keeping track where infinity goes, then compress the three quarters we get the outside sent too to a half plane (again keeping track of infinity), then send that to the unit circle with infinity to 0 and then invert with

z→c/z

 to get the outside circle of right radius and send infinity back to itself, but I didn't manage to get the correct result. Could you help me?

Edited April 27, 2024 by Supercazzola

[Supercazzola]

Someone to help me?

[abuislam]

To get the conformal map, we use a series of transformations. First, use a Möbius transform to map the semicircle to two perpendicular lines, then compress the resulting region to a half-plane. Next, map the half-plane to the unit disk, invert it to get the exterior of a circle, and finally adjust to get the right radius. The resulting map is:

ℎ(𝑢)=𝑖𝑅+𝑢+𝑖𝑅2−𝑢2h(u)=iR+u+iR2−u2

with 𝑢=𝑥1+𝑖𝑥2u=x1+ix2.

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This covers the main steps briefly while addressing the request.