Due to the functional form of typical Lagrangian densities that arise in particle physics, field theorists run into integrals having integrands that are fractions with polynomial denominators when they calculate propagators and Green‘s functions. That is where talk of “poles” and “contour integrals” comes into the picture. Pi makes an entry via the Residue Theorem. Pi shows up in the Residue Theorem because a circular contour surrounding a pole in the complex plane is used as the path of integration of an analytic function. In another thread, Matt Grime said: For some reason, that triggered this thought: Is there any motivation to consider higher dimensional versions of the Residue Theorem? Or would nothing further be gained by this? For instance, would it make sense to consider analytic functions over quaternion values instead of over complex values? Would a quaternion version of the Residue Theorem then result from considering an integral over four-dimensional quaternion space, with the path of integration somehow being a three-dimension volume within the four-dimensional space, analogous to the familiar one-dimensional closed curve within the two-dimensional complex plane? Or does it not even make sense to speak of path integrals where the path is more than one-dimensional? Sorry if this seems kind of fuzzy and ill-defined. At the moment I am not familiar enough with the subject to sharpen my question any further.