Conformal Maps

This page displays a few families of conformal equivalences between subsets of H\mathbb{H} and H\mathbb{H} , parametrized by half plane capacity (the animations themselves, however, are parametrized by the square root of the half plane capacity, since hcap(rA)=r2hcap(A)\mathrm{hcap}(rA) = r^2\, \mathrm{hcap}(A) ).

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This family of conformal transformations is parametrized by the function gt(z)=z2+4t,g_t(z) = \sqrt{z^2 + 4t}, which is the unique conformal transformation that maps H[0,2ti]\mathbb{H} \setminus [0, 2\sqrt{t}i] to H\mathbb{H} such that gt(z)z0 as z.|g_t(z) - z| \rightarrow 0\text{ as } z \rightarrow \infty. The line [0,2ti][0, 2\sqrt{t}i] is displayed in black.

Note that in the square root, we pick the non-negative real axis to the branch cut, and so the function is not defined on z=iyz = iy for y<2tiy \lt 2\sqrt{t} i . In particular, the conformal map does not extend to the origin, hence we see that the map is not conformal "at the origin". Instead, it flattens out the right angle to a horizontal line.

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This is the family of conformal transformations given by gt(z)=z+tz,g_t(z) = z + \frac{t}{z}, which is the unique conformal transformation that maps HtD\mathbb{H} \setminus \sqrt{t}\mathbb{D} to H\mathbb{H} such that g(z)z0 as z.|g(z) - z| \rightarrow 0\text{ as } z \rightarrow \infty. The region tD\sqrt{t} \mathbb{D} is colored in green.