Final answer:
The Cauchy integral formula results in zero when the function integrated has no poles within the curve or if the function is analytic within the enclosed region. In a physical context, if the function is zero outside a certain interval, the integrals over those regions outside will also be zero.
Step-by-step explanation:
The question relates to a situation in mathematics where you might expect the Cauchy integral formula to give a value of zero. The Cauchy integral formula is a key result in complex analysis and is used to evaluate integrals of analytic functions over closed curves. Essentially, you would obtain a value of zero if the function being integrated does not have a pole (a point of undefined value that the function approaches) within the region enclosed by the curve over which you are integrating, or if the function is analytic (differentiable) throughout the closed curve including the interior. For a Cauchy integral set up as an integral from negative infinity to zero, zero to L, and L to infinity, one typical scenario where the integral yields zero is when the function being integrated, commonly symbolized as C, is zero at the limits of integration. This situation occurs when the function C represents a physical constraint, like a particle constrained in a tube, such that C equals zero outside of the region [0, L], hence the integrals from negative infinity to zero and from L to infinity would both yield zero.