Quotient of Entire Functions

w as a quotient: Any function meromorphic on the complex plane can be written as a quotient of two functions that are analytic on the entire plane: entire functions. In the case of our w functions, there is a neat way to find these functions. If we divide our differential equation by w², we can rewrite it as (zw'/w)' = aw⁻² − bw². Note that the left side can also be written (z (ln w)')', so if w = p/q, the equation becomes (z(ln p)')' − (z(ln q)')' = aq²/p² − bp²/q², or (zpp" + pp' − zp'²)/p² − (zqq" + qq' − zq'²)/q² = aq²/p² − bp²/q².

The last suggests pairing terms with the same denominator, and looking at the separate equations zpp" + pp' − zp'² = aq² and zqq" + qq' − zq'² = bp². In fact, if w is everywhere meromorphic, then it is possible to find p and q so these two equations hold; and they are unique if we assume p(0) = q(0) = 1. Indeed, we know there are entire functions f, g, with no common zeros, with f/g = w, and it follows that (zff" + ff' − zf'²)/f² − ag²/f² = (zgg" + gg' − zg'²)/g² − bf²/g². Both sides are meromorphic, and have no common poles. Since they are equal, neither can have any poles, and they both represent the same holomorphic function h. We can solve (zkk" + kk' − zk'²)/k² = h by taking the integral of h (with value 0 at z - 0), dividing by z, integrating again, and then exponentiating. Because of the exponentiation, k has no zeros, so p = f/k and q = g/k do the job.

The functions p and q: If we write p_a,b and q_a,b for the solutions with given a and b, it is clear from the symmetry of the equations, that p_a,b = q_b,a. The two equations can be solved simultaneously in series form. The coefficient of each power of z is a homogeneous polynomial in a and b, of which I calculated over 100 to check various formulas. But I never got a general one. One interesting formula is that the sum of the coefficients of the nth coefficient polynomial is 1/n!, which follows from the fact that the solution when a = b = 1 is p = q = e^z. More obvious is that all coefficients appear to be positive. Added to the previous fact, this would imply that p and q are entire and give an alternate proof that w is everywhere meromorphic.

Graphs: Minor modification of the program for graphing w gives graphs of p and q. Their growth appears to be fairly uniformly exponential in all direction, except for the zeroes, from which the growth appears to recover. The number of zeroes appears to grow proportionately to the distance from the origin. If you would like to do some interactive graphing, you can do so either for graphs of w, p, and q or of p, q, p', q', p", and q".

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