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.

Read more about this topic:  Power Series

Famous quotes containing the words power, series and/or variables:

    “My most desire my hand may reach,
    My will is alway at my hand;
    Me need not long for to beseech
    Her that hath power me to command.”
    —Sir Thomas Wyatt (1503?–1542)

    “Galileo, with an operaglass, discovered a more splendid series of celestial phenomena than anyone since.”
    —Ralph Waldo Emerson (1803–1882)

    “The variables are surprisingly few.... One can whip or be whipped; one can eat excrement or quaff urine; mouth and private part can be meet in this or that commerce. After which there is the gray of morning and the sour knowledge that things have remained fairly generally the same since man first met goat and woman.”
    —George Steiner (b. 1929)