Meromorphic function, Weierstrass product

From: Tony (Ttiger222_at_hotmail.com)
Date: 01/05/05 

Date: Wed, 5 Jan 2005 14:15:32 -0500

Dear all,

I have several questions about the following 2 problem :

Find a meromorphic function on the complex plane having simple poles at
1,2,3,4,... and no other poles.

Well, the Weierstrass infinite product

Product_1^infinity E_1(z/n) = Product_1^infinity (1-z/n)*e^(z/n)

has simple zeroes at n = 1,2,3,4,..

My first reaction to find a meromorphic function would be to take 1/f. But
when I saw a solution, the function taken was f ' / f. Why is this?

Secondly, I am looking for a function meromorphic on the complex plane with
a pole at ni for n = 1,2,3,.... with principal part

sqrt(n) / (z-ni) for each positive integer n and with no other poles.

I can take again

Product_1^infinity E_1(z/ni) = Product_1^infinity (1-z/ni)*e^(z/ni) as my
function that has simple zeroes at i, 2i, 3i, ...

Now I take 1/f (or f ' / f, if that is correct) . How will I know what the
principal part is, and if it isn't what I am looking for, how do I get a
function with a specific principal part?

Thanks for any help,

Tony

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