Causality and response functions Referring to David Tong's notes on Electromagnetism, page 29 (of the PDF, numbered 183), section 7.5.4;
It is proved that the frequency domain response function (in this case describing the polarisation from an applied electric field, but more generally in any linear time/space invariant system) $\alpha(\omega)$ is analytic in the upper half plane, starting with the requirement of causality.
I follow all of the reasoning, up to the point where David says "By the residue theorem, the integral is just given by the sum of residues inside the contour. If we want $\alpha(t) = 0$ for $t < 0$, we need there to be no poles".
When I get to this point, to me it says only that we need the sum of all of the residues at all of the poles inside this contour to be equal to zero. To infer that this means that we have no poles means then there must be some theorem which says there is no way in which all of the residues at these poles could add to give zero. Is this true?
If no such theorem exists, I propose a function which has two poles for which the residue of the first pole is minus the residue at the second pole. I will postpone looking for a specific example of such a function until someone answers my question, in case I end up wasting a large amount of time.
 A: Hopefully the following gives an idea of a proper derivation of the
properties you are interested in. 
Suppose that the polarisation is a linear function of the applied electric
field
\begin{equation*}
\mathbf{P}(\mathbf{x},t)=\int_{\mathbb{R}^{3}}d\mathbf{y}\int_{-\infty
}^{+\infty }ds\chi (\mathbf{x,y},t,s)\mathbf{E}(\mathbf{y},s)
\end{equation*}
In the optical regime the wavelength of the field is much larger than
characteristic distances of the material and the above reduces to
\begin{equation*}
\mathbf{P}(\mathbf{x},t)=\int_{-\infty }^{+\infty }ds\chi (\mathbf{x,}t,s)
\mathbf{E}(\mathbf{x},s)
\end{equation*}
Suppose that the field is switched on at a finite time, say $t=0$, at which
time the polarisation is vanishing (electrets are excluded). Then $\mathbf{P}
(\mathbf{x},t)=0$ for $t\leqslant 0$. Causality requires that $\mathbf{P}(
\mathbf{x},t)$ can only depend on the field $\mathbf{E}(\mathbf{x},s)$ for $
s\leqslant t$ and we end up with
\begin{equation*}
\mathbf{P}(\mathbf{x},t)=\int_{0}^{t}ds\chi (\mathbf{x,}t,s)\mathbf{E}(%
\mathbf{x},s),\;t\geqslant 0
\end{equation*}
Finally homogeneity in time (we can shift everything over a time interval
without change) leads to
\begin{equation*}
\mathbf{P}(\mathbf{x},t)=\int_{0}^{t}ds\chi (\mathbf{x,}t-s)\mathbf{E}(
\mathbf{x},s)=\int_{0}^{t}du\chi (\mathbf{x,}u)\mathbf{E}(\mathbf{x}
,t-u),\;t\geqslant 0
\end{equation*}
In view of the above we can assume that $\chi (\mathbf{x,}t)$ vanishes for $
t<0$. Taking the $\mathbf{x}$ dependence for understood we can take complex
Laplace transforms according to
\begin{eqnarray*}
\hat{f}(z) &=&\int_{0}^{\infty }dt\exp [izt]f(t),\;Imz>0 \\
f(t) &=&\frac{1}{2\pi }\int_{\Gamma }dz\exp [-izt]\hat{f}(z),\;t\geqslant 0
\end{eqnarray*}
where $\Gamma $ is a straight line parallel to and above the real axis. Then
\begin{equation*}
\mathbf{\hat{P}}(z)=\hat{\chi}(z)\mathbf{\hat{E}}(z)
\end{equation*}
Here
\begin{equation*}
\hat{\chi}(z)=\int_{0}^{\infty }dt\exp [izt]\chi (t),\;Imz>0
\end{equation*}
and has a limit as $Imz\downarrow 0$ provided $\chi (t)$ is
absolutely integrable. We can now obtain Kramers-Kronig relations if so
desired.
We now come to our next requirement, passivity. This means that the system
can dissipate but not gain energy. This is discussed in A. Tip, Phys Rev
E69, 016610, 2004. As a result
\begin{equation*}
Imz\hat{\chi}(z)\geqslant 0,\;Imz>0.
\end{equation*}
Provided a further technicality, true if $\chi (0)=0$, holds this makes $%
\hat{\chi}(z)$ a Herglotz function (for a convenient discussion, see R.
Carmona and J. Lacroix, Spectral theory of random Schrödinger operators,
Birkhäuser, Boston 1990) and has the representation
\begin{equation*}
\hat{\chi}(z)=\int \sigma (d\lambda )\frac{1}{\lambda -z}
\end{equation*}
where $\sigma $ is a non-negative finite measure. In general it decomposes
into a sum of point measures, an absolutely continuous measure and singular
continuous measure. The last is usually absent, point measures describe
dispersive, non-absorptive systems and absolutely continuous measures
absorptive systems.
