On differential equations for the Feynman integral of a one-loop diagram Journal Theoretical and Mathematical Physics

The Feynman integral $I(s,t)$ for one-loop diagram with four vertices is considered. With the aid of the Griffiths' method of differentiating rational differential forms with respect to the parameter, it is proved that $I(s,t)$ satisfies the system of two first order differential equations. From this system a hyperbolic partial differential equation for $I(s,t)$ is obtained, the main coefiicient of which vanishes on the Landau's manifold of the Feynman integral.