so I only need to determine countably many numbers, yay
($\sum_{n \in \Bbb Z} a_n z^n$)
wait this is like Fourier series except they're not orthogonal
good grief, what is the explicit mapping
wait a Mobius transformation should do it
$T$ from punctured unit disc to upper half plane
b/d=i, (a-b)/(c-d)=-1, (a+b)/(c+d)=1
b=i, d=1, a=c+1-i, c+1-2i=1-c, c=i, a=1
$T(z) = \dfrac{z+i}{iz+1}$
the inverse is $\dfrac{z-i}{-iz+1}$
so the function on the upper half plane is $\sum_{n\in\Bbb Z} a_n \left(\dfrac{z-i}{-iz+1}\right)^n$
