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MittagLeffler and weierstrass theorem 
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#1
Jul2009, 09:09 PM

P: 9

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 MittagLeffler 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. 


#2
Jul2209, 11:33 AM

P: 392




#3
Jul2309, 04:50 PM

P: 9

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. 


#4
Jul2309, 10:18 PM

P: 392

MittagLeffler and weierstrass theorem
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. 


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