Power Series - Power Series in Several Variables

Power Series in Several Variables

An extension of the theory is necessary for the purposes of multivariable calculus. A power series is here defined to be an infinite series of the form

f(x_1,\dots,x_n) = \sum_{j_1,\dots,j_n = 0}^{\infty}a_{j_1,\dots,j_n} \prod_{k=1}^n \left(x_k - c_k \right)^{j_k},

where j = (j1, ..., jn) is a vector of natural numbers, the coefficients a(j1,...,jn) are usually real or complex numbers, and the center c = (c1, ..., cn) and argument x = (x1, ..., xn) are usually real or complex vectors. In the more convenient multi-index notation this can be written

f(x) = \sum_{\alpha \in \mathbb{N}^n} a_{\alpha} \left(x - c \right)^{\alpha}.

The theory of such series is trickier than for single-variable series, with more complicated regions of convergence. For instance, the power series is absolutely convergent in the set between two hyperbolas. (This is an example of a log-convex set, in the sense that the set of points, where lies in the above region, is a convex set. More generally, one can show that when c=0, the interior of the region of absolute convergence is always a log-convex set in this sense.) On the other hand, in the interior of this region of convergence one may differentiate and integrate under the series sign, just as one may with ordinary power series.

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