Through the trick of introducing the variable v = w"/w we can solve the difficulty of singularity at zeros of w, and by flipping to 1/w, similarly deal with poles of w. We have done nothing to deal with the fact that z = 0 is usually a singularity except to work with solutions that happened to be regular there. To handle the more general case, we can replace w with u(z) = w(e^z), which satisfies the equation uu" − u'² = e^z(1 − u⁴). You can see graphs of this on various domains and with various starting values at z = 1. Note: Although the graph is laid out like u, the program is really calculating w, and those are the starting values requested. For "log radius", input r > 0 such that the z values used are e^-r < |z| < e^r and the angle θ requested is such that -θ < arg z < θ.

you will generally notice strips of zeros and strips of poles, separated by curves where |u| = 1. These strips tend to slant downward (toward z = 0) as arg z moves away from 0, but if w and w' are chosen real, the strips will appear horizontal. The horizontal strips were the first hint that these real solutions have a special property of being almost periodic. If a solution is followed around the unit circle (or any circle centered at the origin), then any solution regular at 0 will return to the same w, w', but most solutions will not. However, many of them will return after a sufficient number of circuits, and all real solutions will come arbitrarily close infinitely often.

Here is a sketch of the proof:

Rewrite the equation in terms of u(z) = ln(w(e^z)). The new equation is u" = -2e^zsinh(2u) = e^x-2u − e^x+2u. From the first expression, each real solution is concave down when u is positive and up when u is negative. From the second expression, u" is approximately -e^x+2u if u >> 0 and e^x-2u if u << 0.

If |u'| > |x|/2 as x approaches -∞ then u" will grow large and cause u' to change sign. Further, since in terms of t = x ± 2y, u" = -e^t gives symmetric solutions, u will cross lines x = ± 2y + c at opposite angles, meaning that u' changes to 1 - u', so that u' moves one unit closer to 0 as long as it is larger than 1, and if |u'| is between 1/2 and 1, then at the next step it is between 0 and 1, and stabilizes.

Once u' takes on a (virtually) constant value r, we can see the near periodicity of the original w by taking a sequence p_n/q_n (from the continued fraction for r, for example), and following the solution along the vertical line from x to x + 2q_nπi. Since this changes u by (virtually) the constant amount 2q_nrπi, and q_nr is (virtually) an integer, w = e^u is (virtually) unchanged.

My conjecture is that the behavior of these solutions with respect to r is due to the solution being of the form f(z, z^r), where f is an analytic function of two complex variables. Moreover, I conjecture that every solution is of this form for some complex r with −1/2 ≤ Re(r) ≤ 1/2, and that every such complex number occurs except ±1/2.