the Hessian matrix. Applying the multi-index notation the Taylor series for several variables becomes which is to be understood as a still more abbreviated multi-index version of the first equation of this paragraph, with a full analogy to the single variable case. Example In order to compute a second-order Taylor series expansion around point of the function one first computes all the necessary partial derivatives: Evaluating these derivatives at the origin gives the Taylor coefficients Substituting these values in to the general formula produces Since is analytic in , we have Comparison with Fourier series The trigonometric Fourier series enables one to express a periodic function (or a function defined on a closed interval ) as an infinite sum of trigonometric functions (sines and cosines). In this sense, the Fourier series is analogous to Taylor series, since the latter allows one to express a function as an infinite sum of powers. Nevertheless, the two series differ from each other in several relevant issues: The finite truncations of the Taylor series of about the point are all exactly equal to at . In contrast, the Fourier series is computed by integrating over an entire interval, so there is generally no such point where all the finite truncations of the series are exact. The computation of Taylor series requires the knowledge of the function on an arbitrary small neighbourhood of a point, whereas the computation of the Fourier series requires knowing the function on its whole domain interval. In a certain sense one could say that the Taylor series is "local" and the Fourier series is "global". The Taylor series is defined for a function which has infinitely many derivatives at a single point, whereas the Fourier series is defined for any integrable function. In particular, the function could be nowhere differentiable. (For example, could be a Weierstrass function.) The convergence of both series has very different properties. Even if the Taylor series has positive convergence radius, the resulting series may not coincide with the function; but if the function is analytic then the series converges pointwise to the function, and uniformly on every compact subset of the convergence interval. Concerning the Fourier series, if the function is square-integrable then the series converges in quadratic mean, but additional requirements are needed to ensure the pointwise or uniform convergence (for instance, if the function is periodic and of class C1 then the convergence