Vertex Function in Quantized Field Theories

An integral representation is given for the vertex function $F(k^2, p^2, {\mathrm{}(k-p)\mathrm{}}^2)$. This representation is obtained on the basis of local commutativity and the spectral conditions. It exhibits the set of points $k^2$, $p^2$, for which $F$ is analytic in ${\mathrm{}(k-p)\mathrm{}}^2$ except for the physical cut. The limitations found in the representation are discussed on the basis of examples obtained from perturbation theory. These examples give some insight into the question of further analytic continuation in $k^2$ and $p^2$.