syms k l m
% l=zeta1, m=zeta2
[ph, r] = meshgrid((0:5:360)*pi/180,[0:.05:0.5, 1:.5:10]);
[x, y] = pol2cart(ph,r);
Z = x + 1i*y;
J = besselj(k,l.*r);
J2 = besselj(k,m.*r);
Y = bessely(k,l.*r);
Y2 = bessely(k,m.*r);
H = besselh(k,r);
F1 = symsum((J).*exp(1i*k*ph),k,-5,5);
%F1 = symsum((J+Y).*exp(1i*k*ph),k,-5,5);
F2 = symsum((J2+Y2).*exp(1i.*k.*ph),k,-5,5);
F3 = symsum(H.*exp(1i.*k.*ph),k,-5,5);
pwu = nan(size(F1), 'like', F1);
mask = 0 <= r & r < 0.5;
pwu(mask) = F1(mask);
mask = 0.5 <= r & r < 1;
pwu(mask) = F2(mask);
mask = r >= 1;
pwu(mask) = F3(mask);
U = subs(pwu, {l, m}, {1.5, 3});
surf(x, y, real(double(U)))
hold on
surf(x, y, zeros(size(x)))
hold off
You created Y but you did not use it. By parallelism with F2 it looks like you would want (J+Y) for F1 instead of just J. I show that version in a comment. The difference is fairly noticable, because at r = 0, bessly(k,0) is infinite, and you get infinities and NaN showing up.
Your boundary conditions were broken: you excluded the cases where r was exactly the boundaries, but those were important cases.
