Editing Power series (section)

== Analytic functions ==
{{main|Analytic function}}
A function ''f'' defined on some [[open set|open subset]] ''U'' of '''R''' or '''C''' is called [[Analytic function|analytic]] if it is locally given by a convergent power series. This means that every ''a'' ∈ ''U'' has an open [[neighborhood (topology)|neighborhood]] ''V'' ⊆ ''U'', such that there exists a power series with center ''a'' that converges to ''f''(''x'') for every ''x'' ∈ ''V''.

Every power series with a positive radius of convergence is analytic on the [[topological interior|interior]] of its region of convergence. All [[holomorphic function]]s 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 ''a''<sub>''n''</sub> can be computed as
<math display="block">a_n = \frac{f^{\left( n \right)} \left( c \right)}{n!}</math>

where <math>f^{(n)}(c)</math> denotes the ''n''th derivative of ''f'' at ''c'', and <math>f^{(0)}(c) = f(c)</math>. 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 [[connectedness|connected]] open set ''U'', and if there exists an element {{math|''c'' ∈ ''U''}} such that {{math|1=''f''{{i sup|(''n'')}}(''c'') = ''g''{{i sup|(''n'')}}(''c'')}} for all {{math|''n'' ≥ 0}}, then {{math|1=''f''(''x'') = ''g''(''x'')}} for all {{math|''x'' ∈ ''U''}}.

If a power series with radius of convergence ''r'' is given, one can consider [[analytic continuation]]s of the series, that is, analytic functions ''f'' which are defined on larger sets than {{math|{{mset| ''x'' | {{abs|''x'' − ''c''}} < ''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]] {{mvar|x}} with {{math|1={{abs|''x'' − ''c''}} = ''r''}} such that no analytic continuation of the series can be defined at {{mvar|x}}.

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

=== Behavior near the boundary ===

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:

# ''Divergence while the sum extends to an analytic function'': <math display="inline">\sum_{n=0}^{\infty}z^n</math> has radius of convergence equal to <math>1</math> and diverges at every point of <math>|z|=1</math>. Nevertheless, the sum in <math>|z|<1</math> is <math display="inline">\frac{1}{1-z}</math>, which is analytic at every point of the plane except for <math>z=1</math>.
# ''Convergent at some points divergent at others'': <math display="inline">\sum_{n=1}^{\infty}\frac{z^n}{n}</math> has radius of convergence <math>1</math>. It converges for <math>z=-1</math>, while it diverges for <math>z=1</math>.
# ''Absolute convergence at every point of the boundary'': <math display="inline">\sum_{n=1}^{\infty}\frac{z^n}{n^2}</math> has radius of convergence <math>1</math>, while it converges absolutely, and uniformly, at every point of <math>|z|=1</math> due to [[Weierstrass M-test]] applied with the [[Harmonic series (mathematics)#p-series|hyper-harmonic convergent series]] <math display="inline">\sum_{n=1}^{\infty}\frac{1}{n^2}</math>.
# ''Convergent on the closure of the disc of convergence but not continuous sum'': [[Wacław Sierpiński|Sierpiński]] gave an example<ref>{{cite journal|author=Wacław Sierpiński|title=Sur une série potentielle qui, étant convergente en tout point de son cercle de convergence, représente sur ce cercle une fonction discontinue. (French)|journal=Rendiconti del Circolo Matematico di Palermo| url=https://zbmath.org/?q=an:46.1466.03|year=1916|volume=41|publisher=Palermo Rend.|pages=187–190 | doi=10.1007/BF03018294 |jfm=46.1466.03 | s2cid=121218640| author-link=Wacław Sierpiński}}</ref> of a power series with radius of convergence <math>1</math>, convergent at all points with <math>|z|=1</math>, but the sum is an unbounded function and, in particular, discontinuous. A sufficient condition for one-sided continuity at a boundary point is given by [[Abel's theorem]].