Prove that a hyperbolic circle with hyperbolic radius in the hyperbolic upper half plane is
The problem suggests to place a coordinate system at the Euclidean of the circle and work out the double integral in polar coordinates. However, and I believe I must be messing up somewhere, if we let the Euclidean center of the circle be and have Euclidean radius , then we have in polar coordinates centered at the Euclidean center and , and so we have the hyperbolic area of the circle being:
However, this integral seems to be undefined because along the line somewhere, I get a , which is undefined at and .
Are my parameterizations wrong? Or are they correct and this integral does exist? Should I not deal with the Euclidean radius and approach it some other way? Thanks.
So my limits of integration are incorrect? I use this identity:
for the first integral (the part) upon which I get:
So this is defined? If so, what does this integral equal? When I use integral calculators, a appears, and therefore the whole expression becomes undefined. So either my limits for integration are incorrect or these integral calculators are all wrong. Thanks.
Yes, it is; you can see that by considering an asymptotic expansion to order 2 when .
But this integral is indeed complicated; maybe you shouldn't use polar coordinates. Using the initial Cartesian coordinates, the area is (letting )
.
This may be easier to compute (at least it looks nicer).