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Two circles lie outside one another except for common point of tangency. How to map the region exterior to both circles (including the point of inifity) onto an infinite strip by one to one comformal mapping?

My first thought was to map the region on to the unit disk, then map it to the strip. But the boundary of two circles are not a generalized circle in Riemann sphere, I can not use Mobius transformation. And other functions like square, exp, rotation and translation are also not a good choice.

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Any Mobius transformation that brings the point where the circles are tangent to the point at infinity will transform the two circles into parallel lines.

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