Are you wanting a nice, smooth map? A bijection? Or just some function?

I can't think of a neat one - I'm sure a neat one will exist! However, assume your annulus is centered at the origin. Then you know that you can take one closed interval [a, b] to another [c, d] by the function . So, generalise this. Take the point in your annulus and simply map it so far along the line (that is, on the positive part of the x-axis you will just be moving the interval to , and just rotate this all the way around).

I will leave you to fill in the details.