Dispersion relations for the vertex function from local commutativity

Dispersion relations for the vertex function are derived which are valid when two of the scalar variables are arbitrary complex inside certain domains of the product of the complex planes and the third scalar variable is evaluated just below or just above the physical region-cut.The domains of validity of the dispersion relations for the complex variables are domains with three real dimensions and can be described as neighbourhoods of the boundaries of the axiomatic analyticity region of Källén and Wightman.The discontinuity of the vertex function across the cut-surface in the third variable for such values of the remaining variables is expressed only in terms of the dynamical on-mass-shell matrix elements of the locally commuting field operators.