a two–point function. The integral runs over all possible values of the euclidean momenta and therefore it requires, a priori, the knowledge of the two–point function at all distances. I want to stress the fact that it is not enough to know the very low $Q^2$ behaviour of the $\Pi_{LR}(Q^2)$ correlation function to obtain an approximation to the r.h.s. integral in eq (\[eq:piem\]). The low $Q^2$ behaviour of this function is governed by chiral perturbation theory and the first two terms are known $$\Pi_{LR}(Q^2)=-\frac{f_{\pi}^2}{Q^2}-4L_{10}+{{\cal O}}(Q^2);$$ but this behaviour cannot be extrapolated to the ultraviolet region of the integral since it would lead to a divergent result. By contrast, as we shall further discuss in section \[subsec:wsrope\], the full $Q^2$ resonance dominance ansatz in eq. (\[eq:lrtpfrd\]) with the Weinberg sum rule constraints in eqs. (\[eq:tsc\]) has both the long–distance behaviour and the short–distance behaviour of QCD.

Dispersion Relations {#sec:disperrels}
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In these lectures we shall be all the time considering two–point functions of local operators i.e., Fourier transforms of the vacuum expectation value of the time ordered product of a local current $J(x)$ times its hermitian conjugate $$\label{eq:tpf}
\Pi(q^2)=i\int d^{4}x e^{iq\cdot x}\langle 0\vert
\mbox{\rm T}\left\{J(x)J(0)^{\dagger}\right\}\vert 0\rangle.$$ In most applications the current 