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
