# Analytic function

An **analytic function** is a function that can be represented locally by a convergent power series. In complex analysis, the term is equivalent to *holomorphic*: a complex function is analytic at a point if it is complex differentiable in a neighborhood of that point, and this differentiability alone forces a full power series expansion.<sup>[1](https://www.math.cuhk.edu.hk/course_builder/1819/math2230b/3.pdf)</sup><sup> • </sup><sup>[2](https://people.math.osu.edu/costin.9/7651-2021/adnotes.pdf)</sup> Many mathematicians prefer the term "holomorphic function," while "analytic" remains widespread among physicists, engineers, and older texts.<sup>[3](https://mathworld.wolfram.com/AnalyticFunction.html)</sup>

| Key fact | Detail |
|---|---|
| Definition | A complex function is analytic (holomorphic) at a point if it is complex differentiable throughout a neighborhood of that point; it is analytic on a domain if it is complex differentiable at every point of the domain<sup>[1](https://www.math.cuhk.edu.hk/course_builder/1819/math2230b/3.pdf)</sup><sup> • </sup><sup>[10](https://dlmf.nist.gov/1.9)</sup> |
| Central equivalence | Analytic functions are exactly those admitting a convergent power series expansion in a neighborhood of every point of their domain<sup>[2](https://people.math.osu.edu/costin.9/7651-2021/adnotes.pdf)</sup> |
| Taylor series | Around each point \( z_0 \), \( f(z)=\sum_{j=0}^{\infty}\frac{f^{(j)}(z_0)}{j!}(z-z_0)^j \), converging in the largest open disk contained in the domain<sup>[4](https://homepages.uc.edu/~herronda/complex_analysis/Texts/Intro2ComplexAnalysis.pdf)</sup> |
| Derivatives | An analytic function has derivatives of all orders, and each derivative is itself analytic<sup>[2](https://people.math.osu.edu/costin.9/7651-2021/adnotes.pdf)</sup> |
| Radius of convergence | The Taylor series radius equals the distance from \( z_0 \) to the nearest singularity of \( f \)<sup>[2](https://people.math.osu.edu/costin.9/7651-2021/adnotes.pdf)</sup> |
| Singularities | At an isolated singularity the Taylor series is replaced by the Laurent series, which includes negative-degree terms<sup>[5](https://math.libretexts.org/Bookshelves/Analysis/Complex_Variables_with_Applications_%28Orloff%29/08%3A_Taylor_and_Laurent_Series)</sup> |
| Continuation | Analytic continuation is ill-posed; even with a boundedness assumption it has an infinite condition number<sup>[6](https://link.springer.com/article/10.1007/s10543-020-00802-7)</sup> |

## Definition and the two notions

For a function of a complex variable, being analytic at a point means it is complex differentiable throughout some neighborhood of that point, and the standard course sequence proves Cauchy's theorem and [Cauchy's integral formula](https://www.edgepedia.org/cauchys-integral-formula) from this starting point.<sup>[5](https://math.libretexts.org/Bookshelves/Analysis/Complex_Variables_with_Applications_%28Orloff%29/08%3A_Taylor_and_Laurent_Series)</sup><sup> • </sup><sup>[10](https://dlmf.nist.gov/1.9)</sup> These results reveal deep properties that differentiability alone would not suggest, such as the existence of derivatives of all orders.<sup>[5](https://math.libretexts.org/Bookshelves/Analysis/Complex_Variables_with_Applications_%28Orloff%29/08%3A_Taylor_and_Laurent_Series)</sup> Riemann proved that a holomorphic function is analytic in the series sense, that is, it has a local convergent series representation.<sup>[7](https://www.cs.purdue.edu/homes/spa/courses/pg17/mybook.pdf)</sup>

The name "analytic" comes from the fact that a holomorphic function can be expanded in a power series locally around every point of its domain.<sup>[8](https://www.math.wustl.edu/~sk/math416.02-03-2022.pdf)</sup>

## The Taylor series and its consequences

**Taylor's theorem.** If \( f \) is analytic in a domain \( S \) and \( B(z_0,R) \) is the largest open disk in \( S \) centered at \( z_0 \), then

\[ f(z)=f(z_0)+\frac{f'(z_0)}{1!}(z-z_0)+\frac{f''(z_0)}{2!}(z-z_0)^2+\cdots=\sum_{j=0}^{\infty}\frac{f^{(j)}(z_0)}{j!}(z-z_0)^j, \]

converging for all \( z \in B(z_0,R) \), and uniformly on every closed disk \( \overline{B}(z_0,r) \) with \( 0 \le r < R \).<sup>[4](https://homepages.uc.edu/~herronda/complex_analysis/Texts/Intro2ComplexAnalysis.pdf)</sup> The proof requires only Cauchy's integral formula for derivatives: the kernel \( 1/(\zeta-z) \) is expanded as a [geometric series](https://www.edgepedia.org/geometric-series) and integrated term by term, giving the coefficients \( f^{(j)}(z_0)/j! \).<sup>[4](https://homepages.uc.edu/~herronda/complex_analysis/Texts/Intro2ComplexAnalysis.pdf)</sup>

This theorem completes a two-way story. A power series converges to an analytic function inside its disk of convergence, and [Taylor's theorem](https://www.edgepedia.org/taylors-theorem) gives the converse: around each point of analyticity an analytic function equals a convergent power series.<sup>[5](https://math.libretexts.org/Bookshelves/Analysis/Complex_Variables_with_Applications_%28Orloff%29/08%3A_Taylor_and_Laurent_Series)</sup> The result is described as the central equivalence of complex analysis: analytic functions are exactly those that admit a convergent power series expansion near every point of their domain.<sup>[2](https://people.math.osu.edu/costin.9/7651-2021/adnotes.pdf)</sup>

Two quantitative facts follow. First, the disk of convergence of a [Taylor series](https://www.edgepedia.org/taylor-series) cannot be arbitrarily small: its radius equals the distance from \( z_0 \) to the nearest singularity of \( f \).<sup>[2](https://people.math.osu.edu/costin.9/7651-2021/adnotes.pdf)</sup> Second, the remainder of the partial sum can be bounded by a Cauchy-type estimate: if \( D_\rho(z_0) \subset D \), then

\[ |E(z,z_0,n)| \le \max_{D_\rho(z_0)}|f| \, \frac{|z-z_0|^{n+1}}{\rho-|z-z_0|}, \]

with an optimal version taking the infimum over admissible radii \( \rho' \le \rho \).<sup>[2](https://people.math.osu.edu/costin.9/7651-2021/adnotes.pdf)</sup>

## Rigidity compared with real functions

In real analysis, differentiability need not imply a power series expansion; holomorphic functions are "rigid," their local behavior completely determined by their power series.<sup>[2](https://people.math.osu.edu/costin.9/7651-2021/adnotes.pdf)</sup>

Rigidity also works globally.

## Singularities and Laurent series

A function is singular at a point where it is not analytic.<sup>[5](https://math.libretexts.org/Bookshelves/Analysis/Complex_Variables_with_Applications_%28Orloff%29/08%3A_Taylor_and_Laurent_Series)</sup> At an isolated singularity the Taylor series is replaced by the [Laurent series](https://www.edgepedia.org/laurent-series), a power series representation that includes terms of negative degree and applies where a Taylor expansion cannot.<sup>[5](https://math.libretexts.org/Bookshelves/Analysis/Complex_Variables_with_Applications_%28Orloff%29/08%3A_Taylor_and_Laurent_Series)</sup> Poles are isolated singularities, and the classification of zeros and poles, residues, and the residue theorem build directly on these expansions.<sup>[5](https://math.libretexts.org/Bookshelves/Analysis/Complex_Variables_with_Applications_%28Orloff%29/08%3A_Taylor_and_Laurent_Series)</sup>

## Analytic continuation

A Taylor series determines the function only inside its disk of convergence, and extending an analytic function beyond it is the problem of analytic continuation. This problem is ill-posed in general; it becomes merely ill-conditioned, though with an infinite condition number, when the function is known to be bounded in a given region of the complex plane.<sup>[6](https://link.springer.com/article/10.1007/s10543-020-00802-7)</sup> The geometry matters: in an annulus, the Hadamard three-circles theorem implies the ill-conditioning is not too severe, which explains the effectiveness of Chebfun and related numerical methods in evaluating analytic functions off their interval of definition; in a strip or channel, accuracy is lost exponentially, at the rate \( \exp(-\pi x/2) \), and the classical Weierstrass chain-of-disks method loses digits faster, at the rate \( \exp(-ex) \).<sup>[6](https://link.springer.com/article/10.1007/s10543-020-00802-7)</sup> On the theoretical side, any algorithm for analytic continuation cannot generally depend on finitely many function values alone without closer inspection of the function itself.<sup>[9](https://arxiv.org/abs/1403.2858)</sup>

Continuation in practice exploits alternative formulas rather than Taylor coefficients, since a formula is a finite object while the coefficients form an infinite set. The [Riemann zeta function](https://www.edgepedia.org/riemann-zeta-function) is the standard example: its [Dirichlet series](https://www.edgepedia.org/dirichlet-series) converges only for \( \operatorname{Re} z > 1 \), but other representations extend \( \zeta(z) \) to the whole complex plane.<sup>[6](https://link.springer.com/article/10.1007/s10543-020-00802-7)</sup>

## References

1. [Complex functions: analyticity and differentiability, MATH2230B, The Chinese University of Hong Kong](https://www.math.cuhk.edu.hk/course_builder/1819/math2230b/3.pdf)
2. [Advanced notes, 7651 Complex Analysis (Taylor series and Cauchy's formula for higher derivatives), Ohio State University](https://people.math.osu.edu/costin.9/7651-2021/adnotes.pdf)
3. [Analytic Function, Wolfram MathWorld](https://mathworld.wolfram.com/AnalyticFunction.html)
4. [R.P. Agarwal et al., An Introduction to Complex Analysis, Lecture 24: Taylor's Series, Springer (2011)](https://homepages.uc.edu/~herronda/complex_analysis/Texts/Intro2ComplexAnalysis.pdf)
5. [Taylor and Laurent Series, Complex Variables with Applications (Orloff), Mathematics LibreTexts](https://math.libretexts.org/Bookshelves/Analysis/Complex_Variables_with_Applications_%28Orloff%29/08%3A_Taylor_and_Laurent_Series)
6. [Quantifying the ill-conditioning of analytic continuation, BIT Numerical Mathematics](https://link.springer.com/article/10.1007/s10543-020-00802-7)
7. [Complex Variables lecture notes, Purdue University](https://www.cs.purdue.edu/homes/spa/courses/pg17/mybook.pdf)
8. [Math 416 lecture notes, Washington University in St. Louis](https://www.math.wustl.edu/~sk/math416.02-03-2022.pdf)
9. [Computational analytic continuation, arXiv:1403.2858](https://arxiv.org/abs/1403.2858)
10. [DLMF: §1.9 Calculus of a Complex Variable ‣ Topics of Discussion ‣ Chapter 1 Algebraic and Analytic Methods](https://dlmf.nist.gov/1.9)

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