- #1
dRic2
Gold Member
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Suppose I have a function
$$f(x) = \lim_{\eta \rightarrow 0} \int_{-\infty}^{\infty} d \zeta \frac {g(\zeta)}{x - \zeta + i \eta}$$
and suppose ##g(\zeta)## is a continuous (maybe even differentiable) function. Can ##f(x)## have complex poles of the form ##a + ib## with ##b## not an infinitesimal ?
Would a similar result hold if, instead of an integral, I have a summation
$$f(x) = \lim_{\eta \rightarrow 0} \sum_{i}^{\infty} \frac {g_i}{x - \zeta_i + i \eta}$$
?
I'm sorry if I'm not writing any ideas, but I don't have any. It has been quite a while since my last analysis exam and I don't really known where to even start. Btw this is non an exercise, it's just something I'm wondering about.
Thanks
Ric
$$f(x) = \lim_{\eta \rightarrow 0} \int_{-\infty}^{\infty} d \zeta \frac {g(\zeta)}{x - \zeta + i \eta}$$
and suppose ##g(\zeta)## is a continuous (maybe even differentiable) function. Can ##f(x)## have complex poles of the form ##a + ib## with ##b## not an infinitesimal ?
Would a similar result hold if, instead of an integral, I have a summation
$$f(x) = \lim_{\eta \rightarrow 0} \sum_{i}^{\infty} \frac {g_i}{x - \zeta_i + i \eta}$$
?
I'm sorry if I'm not writing any ideas, but I don't have any. It has been quite a while since my last analysis exam and I don't really known where to even start. Btw this is non an exercise, it's just something I'm wondering about.
Thanks
Ric