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I am interested in somehow characterizing the zeros of the following function:

$P(z) = \sum_{i=1}^n \frac{\alpha_i}{z+\lambda_i}$,

with $\lambda_1\geq \lambda_2 \geq \ldots \geq \lambda_n=0$ and $\sum_{i=1}^n \alpha_i = 0$.

Furthermore, the values $\alpha_i$ are in general non-integer and may even be complex. The poles $\lambda_i$, however, are all real.

I initially had hoped to use something like the Gauss-Lucas theorem for this, but the conditions on $\alpha_i$ seem to prevent this.

While it would be great to make some strong statements about the zeros, I would be more then satisfied to find conditions guaranteeing that the zeros lie in the left-half of the complex plane.

Any thoughts/suggestions would be great. Thanks!

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    Either you meant "the poles $-\lambda_i$" or the sign in the denominator is wrong.2011-03-02
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    yes, you are correct, the poles are in fact $-\lambda_i$.2011-03-02
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    we [discourage simultaneous cross posts to another forum](http://meta.math.stackexchange.com/questions/1733/what-is-the-proper-way-to-acknowledge-an-answer-posted-on-another-forum-site) of the same question. Your question is also [asked at mathoverflow](http://mathoverflow.net/questions/56613/zeros-of-linear-partial-fractions). Please follow Jeff's suggestion in the linked Meta thread in the future.2011-03-03
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    I'll come back to this later; for now I'll note that the stuff [here](http://dx.doi.org/10.1016/0024-3795(94)00314-9) might help a fair bit with answering your question.2011-05-01

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