Editing Taylor series (section)

==Analytic functions==
{{main|Analytic function}}
[[Image:Exp neg inverse square.svg|upright=1.4|thumb|right|The function {{math|1=<strong style="color:#803300">''e''<sup>(−1/''x''<sup>2</sup>)</sup></strong>}} is not analytic at {{math|1=''x'' {{=}} 0}}: the Taylor series is identically 0, although the function is not.]]
If {{math|''f''&thinsp;(''x'')}} is given by a convergent power series in an open disk centred at {{mvar|b}} in the complex plane (or an interval in the real line), it is said to be [[analytic function|analytic]] in this region.  Thus for {{mvar|x}} in this region, {{mvar|f}} is given by a convergent power series

<math display="block">f(x) = \sum_{n=0}^\infty a_n(x-b)^n.</math>

Differentiating by {{mvar|x}} the above formula {{mvar|n}} times, then setting {{math|''x'' {{=}} ''b''}} gives:

<math display="block">\frac{f^{(n)}(b)}{n!} = a_n</math>

and so the power series expansion agrees with the Taylor series.  Thus a function is analytic in an open disk centered at {{mvar|b}} if and only if its Taylor series converges to the value of the function at each point of the disk.

If {{math|''f''&thinsp;(''x'')}} is equal to the sum of its Taylor series for all {{mvar|x}} in the complex plane, it is called [[entire function|entire]]. The polynomials, [[exponential function]] {{math|''e''<sup>''x''</sup>}}, and the [[trigonometric function]]s sine and cosine, are examples of entire functions. Examples of functions that are not entire include the [[square root]], the [[logarithm]], the [[trigonometric function]] tangent, and its inverse, [[arctan]]. For these functions the Taylor series do not [[convergent series|converge]] if {{mvar|x}} is far from {{mvar|b}}. That is, the Taylor series [[divergent series|diverges]] at {{mvar|x}} if the distance between {{mvar|x}} and {{mvar|b}} is larger than the [[radius of convergence]]. The Taylor series can be used to calculate the value of an entire function at every point, if the value of the function, and of all of its derivatives, are known at a single point.

Uses of the Taylor series for analytic functions include:
# The partial sums (the [[Taylor polynomial]]s) of the series can be used as approximations of the function. These approximations are good if sufficiently many terms are included.
#Differentiation and integration of power series can be performed term by term and is hence particularly easy.
#An [[analytic function]] is uniquely extended to a [[holomorphic function]] on an open disk in the [[complex number|complex plane]]. This makes the machinery of [[complex analysis]] available.
#The (truncated) series can be used to compute function values numerically, (often by recasting the polynomial into the [[Chebyshev form]] and evaluating it with the [[Clenshaw algorithm]]).
#Algebraic operations can be done readily on the power series representation; for instance, [[Euler's formula]] follows from Taylor series expansions for trigonometric and exponential functions. This result is of fundamental importance in such fields as [[harmonic analysis]].
#Approximations using the first few terms of a Taylor series can make otherwise unsolvable problems possible for a restricted domain; this approach is often used in physics.