[math-fun] A remarkable probability distribution on the (perimeter of the) unit circle

This density on the circle shares the property with the plain uniform distribution on the circle that the expected value of z^k is ZERO for each integer k with |k|>1. However, when |k|=1 we (unlike plain uniform) do NOT get zero,

--actually... duh... it occurs to me we can also get all well-behaved densities on the circle |z|=1 which have this "remarkable property" by simply making the density be proportional to A + z + 1/z for any constant A>=2. (Proof: fourier series.) So, my particular density which I constructed in a much more complicated way must be expressible as such a formula (that equality is a remarkable identity, I guess).