Power series

In mathematics, a power series (in one variable) is an infinite series of the form

In many situations, c (the center of the series) is equal to zero, for instance when considering a Maclaurin series. In such cases, the power series takes the simpler form

Beyond their role in mathematical analysis, power series also occur in combinatorics as generating functions (a kind of formal power series) and in electronic engineering (under the name of the Z-transform). The familiar decimal notation for real numbers can also be viewed as an example of a power series, with integer coefficients, but with the argument x fixed at 110. In number theory, the concept of p-adic numbers is also closely related to that of a power series.

The set of the complex numbers such that |xc| < r is called the disc of convergence of the series. The series converges absolutely inside its disc of convergence, and converges uniformly on every compact subset of the disc of convergence.

For |xc| = r, there is no general statement on the convergence of the series. However, Abel's theorem states that if the series is convergent for some value z such that |zc| = r, then the sum of the series for x = z is the limit of the sum of the series for x = c + t (zc) where t is a real variable less than 1 that tends to 1.

When two functions f and g are decomposed into power series around the same center c, the power series of the sum or difference of the functions can be obtained by termwise addition and subtraction. That is, if

The sum of two power series will have, at minimum, a radius of convergence of the smaller of the two radii of convergence of the two series (and it may be higher than either, as seen in the example above).[2]

Both of these series have the same radius of convergence as the original one.

A function f defined on some open subset U of R or C is called analytic if it is locally given by a convergent power series. This means that every aU has an open neighborhood VU, such that there exists a power series with center a that converges to f(x) for every xV.

Every power series with a positive radius of convergence is analytic on the interior of its region of convergence. All holomorphic functions are complex-analytic. Sums and products of analytic functions are analytic, as are quotients as long as the denominator is non-zero.

If a function is analytic, then it is infinitely differentiable, but in the real case the converse is not generally true. For an analytic function, the coefficients an can be computed as

The global form of an analytic function is completely determined by its local behavior in the following sense: if f and g are two analytic functions defined on the same connected open set U, and if there exists an element cU such that f(n)(c) = g(n)(c) for all n ≥ 0, then f(x) = g(x) for all xU.

The power series expansion of the inverse function of an analytic function can be determined using the Lagrange inversion theorem.

The sum of a power series with a positive radius of convergence is an analytic function at every point in the interior of the disc of convergence. However, different behavior can occur at points on the boundary of that disc. For example:

In abstract algebra, one attempts to capture the essence of power series without being restricted to the fields of real and complex numbers, and without the need to talk about convergence. This leads to the concept of formal power series, a concept of great utility in algebraic combinatorics.

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