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where the coefficients an, the center a, and the argument x are usually real or complex numbers. These series usually arise as the Taylor series of some known function; the Taylor series article contains many examples.
A power series will converge for some values of the variable x (at least for x = a) and may diverge for others. It turns out that there is always a number r with 0 ≤ r ≤ ∞ such that the series converges whenever |x − a| < r and diverges whenever |x − a| > r. (For |x - a| = r we cannot make any general statement.) The number r is called the radius of convergence of the power series; in general it is given as
(see lim inf) but a fast way to compute it is
The latter formula is valid only if the limit exists, while the former formula can always be used.
The series converges absolutely for |x - a| < r and converges uniformly on every compact subset of {x : |x − a| < r}.
When two functions f and g are decomposed into power series, the power series of the sum or difference of the functions can be obtained by termwise addition and subtraction. That is, if:
then
With the same definitions above, for the power series of the product and quotient of the functions can be obtained as follows:
For division, observe:
and then use the above, comparing coefficients
Once a function is given as a power series, it is continuous wherever it converges and is differentiable on the interior of this set. It can be differentiated and integratedThis article deals with the concept of an integral in mathematical calculus. For other meanings of "integral" see integration. In calculus, the integral of a function is a generalization of area, mass, volume, sum, and total. Unlike the process of differe quite easily, by treating every term separately:
Both of these series have the same radius of convergence as the original one.
A function f defined on some open subsetIn topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can "wiggle" or "change" any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U of R or C is called analytic if it is locally given by power series. This means that every a ∈ U has an open neighborhood V ⊆ U, such that there exists a power series with center a which converges to f(x) for every x ∈ V.
Every power series with a positive radius of convergence is analytic on the interior of its region of convergence. All holomorphic functionHolomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. This is a much stronger condition than rs 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 often differentiable, but in the real case the converse is not generally true. For an analytic function, the coefficients an can be computed as
where f (n)(a) denotes the n-th derivative of f at a. This means that every analytic function is locally represented by its Taylor series.
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 a∈U such that f (n)(a) = g (n)(a) for all n ≥ 0, then f(x) = g(x) for all x ∈ U.
If a power series with radius of convergence r is given, one can consider analytic continuations of the series, i.e. analytic functions f which are defined on larger sets than { x : |x - a| < r } and agree with the given power series on this set. The number r is maximal in the following sense: there always exists a complex number x with |x - a| = r such that no analytic continuation of the series can be defined at x.
The power series expansion of the inverse function of an analytic function can be determined using the Lagrange inversion theorem.
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