Mittag-Leffler and weierstrass theorem

jian1

1. The problem statement, all variables and given/known data

Let G be a region and let {a_n} and {b_m} be two sequences of distinct points in G such that a_n != b_m for all n,m. Let S_n(z) be a singular part at a_n and let p_m be a positive integer. Show that there is a meromorphic function f on G whose only poles and zeros are {a_n} and {b_m} respectively, the singular part at z=a_n is S_n(z) and z=b_m is a zero of multiplicity p_m.

2. Relevant equations



3. The attempt at a solution

I understand that one can construct a function by Mittag-Leffler Theorem to have all these poles and singular parts, and I also know that by Weierstrass theorem, constructing a function with zeros is no problemo. The question is how do you construct a function with all the singular parts, poles and zeros?

My attempts don't work well despite the poles and zeros are preserved, the singular parts are not.

Billy Bob

I'm not sure, but I have an idea. But first, how did you find a function with the desired poles and zeros (just not necessarily the desired singular parts)?

jian1

say you have a function f(z) with all the desired poles, and another function g(z) with desired zeros, since both poles are zeros are distinct, a function f(z)g(z) will be a meromorphic function with both desired poles and zeros.

Now the problem is that singular part(s) in f(z) is no longer preserved in the new function f(z)g(z). and I tried various ways, but neither works.

Billy Bob

Good first attempt. Now one other problem is that f(z) may have zeros, so also divide by h(z) which has the same zeros as f(z), with the correct multiplicity.

So f(z)g(z)/h(z) is almost right. You also want g(z)/h(z) to be = 1 at the poles of f(z). Maybe construct f first, then h, then g.

Now look at an example pole z=a with desired singular part S(z). Near z=a you have f(z) = S(z) + analytic fcn, i.e. f(z)-S(z)=analytic at z=a. You want f(z)g(z)/h(z) - S(z) to be analytic at z=a. Experiment around with those hints. I think there is a further condition on the way you choose g(z) to make that happen, but I don't want to completely give it away without you trying. You might come up with a better way, anyway.