The coefficients of a Taylor series of a function about a given point are related to the nth derivatives of the function at that point. Can we make a similar statement about what the (negative-index) coefficients of the Laurent series of a function mean, perhaps relating to the "nth antiderivative" or something?
I'm aware that all the coefficients of the Laurent series can be expressed using contour integrals, but this is unsatisfying for two reasons. First of all, the coefficients of a Taylor series of a function in the complex plane can also be expressed in terms of contour integrals (because the Taylor series is part of the Laurent series), but that doesn't stop them from being expressed in another way, in terms of nth derivatives. And second of all, the contour integrals involved in the Laurent series expansion are invariant under changes in the contour used (as long as you stay within the annulus of convergence), which suggests to me that the coefficients are based on a deeper property of the function than the mere value of a given contour integral, specifically some property of the function at or in the neighborhood of the point that we're taking the Laurent series about. So what is this property?
Any help would be greatly appreciated.
Thank You in Advance.