thank you very much for creating this website and allow people to ask questions here. This is my question, I want to know if it is possible to find a conformal transformation that maps the inside of the unit circles to their respective line segments as follows: There are 2 circles in the z-plane, both circles have a unitary radius. The first circle is centered at the origin of the z-plane (0,0). The second circle is placed bellow the first circle at a distance of 2ih.The variable h is a real and positive number, is a constant and can be any number. The second circle is centered at the point (0,-2ih).I want to know if it is possible to find a conformal transformation that maps the inside of the first unit circle to a line segment v=0 between u=-1 and u=1 in the w-plane; and the second circle is mapped to its respective line segment v=-2ih' between u-1 and u=1. The variable h' is another variable that represents any constant real and positive number, however it can be different or the same value as h. I have been working in this problem for a long time, but my knowledge of conformal mapping is not so good, I would appreciate any help, Thank you very much.
There is no such thing. The image of an open set under a conformal map is a also open. It cannot be a line segment.
I guess what you actually wanted was to conformally map the exterior of two circle onto the exterior of two line segment. This is possible if and only if the circles are disjoint, that is $h>1$. Indeed, the conformal maps preserve the number of boundary components. If the circles intersect, there is just one boundary component, while the target domain has two.
Suppose $h>1$. Two doubly-connected domains are conformally equivalent if and only if they have the same modulus: see Conformal maps of doubly connected regions to annuli. Your target domain has a parameter $h'$, with which its modulus can be adjusted to match the modulus of the exterior of two circles. Indeed, as $h'\to 0$ the modulus tends to $0$, and as $h'\to\infty$ it tends to $\infty$. By the intermediate value theorem, there is $h'$ for which the exterior of two segments has the same modulus as the exterior of two circles.
Unfortunately, there does not appear to be a nice formula for a conformal map between these domains. Your best chance is working with a Schwarz-Christoffel integral, which becomes more complicated for doubly connected domains, but can still be used: see Schwarz-Christoffel Mapping of Multiply Connected Domains.