Between now and implementation (and to see whether implementation is worth it): The pictures on the given page are the results of a single transformation of his "8 Sector Map" by the given function (a common technique to draw complex functions of a complex variable, since as a graph it would require four dimensions). For initial experiments, note that Bessel functions of half-integer order are elementary: K (z) = exp(-z)*sqrt(pi/2)*(1+1/z)/sqrt(z) 3/2 Mess with that in a formula to see what one can get. To construct some more, note: J (z) = sqrt(2/(pi*z))sin(z-n*pi/2)*W(n,z)+cos(z-n*pi/2)*V(n,z)/sqrt(2*pi*z^3) n+1/2 where n is an integer, floor(n/2) W(n,z) = SUM ((n+2j)!/((2j)!(n-2j)!)(-1/(4z^2))^j j=0 and floor((n-1)/2) V(n,z) = SUM ((n+2j+1)!/((2j+1)!(n-2j-1)!)(-1/(4z^2))^j j=0