Editing Power series (section)

== 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
<math display="block">f(x_1, \dots, x_n) = \sum_{j_1, \dots, j_n = 0}^\infty a_{j_1, \dots, j_n} \prod_{k=1}^n (x_k - c_k)^{j_k},</math>
where {{math|1=''j'' = (''j''<sub>1</sub>, …, ''j''<sub>''n''</sub>)}} is a vector of natural numbers, the coefficients {{math|''a''<sub>(''j''<sub>1</sub>, …, ''j''<sub>''n''</sub>)</sub>}} are usually real or complex numbers, and the center {{math|1=''c'' = (''c''<sub>1</sub>, …, ''c''<sub>''n''</sub>)}} and argument {{math|1=''x'' = (''x''<sub>1</sub>, …, ''x''<sub>''n''</sub>)}} are usually real or complex vectors. The symbol <math>\Pi</math> is the [[multiplication#Capital Pi notation|product symbol]], denoting multiplication. In the more convenient [[multi-index]] notation this can be written
<math display="block">f(x) = \sum_{\alpha \in \N^n} a_\alpha (x - c)^\alpha.</math>
where <math>\N</math> is the set of [[natural number]]s, and so <math>\N^n</math> is the set of ordered ''n''-[[tuple]]s of natural numbers.

The theory of such series is trickier than for single-variable series, with more complicated regions of convergence. For instance, the power series <math display="inline">\sum_{n=0}^\infty x_1^n x_2^n</math> is absolutely convergent in the set <math>\{ (x_1, x_2): |x_1 x_2| < 1\}</math> between two hyperbolas.  (This is an example of a ''log-convex set'', in the sense that the set of points <math>(\log |x_1|, \log |x_2|)</math>, where <math>(x_1, x_2)</math> 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.<ref>{{cite journal |doi=10.1090/S0002-9904-1948-08994-7|title=Convex functions|year=1948|last1=Beckenbach|first1=E. F.|journal=Bulletin of the American Mathematical Society|volume=54|issue=5|pages=439–460|doi-access=free}}</ref>