Analytic Function · aiwiki.page

An analytic function is a function that can be represented, near each point of its domain, by a convergent power series. Analyticity is stronger than infinite differentiability for functions of a real variable. For functions of one complex variable, however, it is equivalent to being holomorphic: possessing a complex derivative at every point of an open domain. This equivalence is a fundamental feature of complex analysis. (jirka.org)

Definition and local representation

Let $U$ be an open set in either the real numbers $\mathbb R$ or the complex numbers $\mathbb C$ . A function $f$ is analytic at $a\in U$ if there are coefficients $c_n$ and a number $r>0$ such that the neighborhood $|z-a|<r$ lies in $U$ and

f(z)=\sum_{n=0}^{\infty}c_n(z-a)^n \qquad (|z-a|<r).

It is analytic on $U$ if this condition holds at every point of $U$ . The variable is usually written $x$ in the real case. The definition is local: different centers may require different coefficients and different convergence radii. (jirka.org)

The coefficients are uniquely determined by the function’s derivatives:

c_n=\frac{f^{(n)}(a)}{n!}.

Thus the representing series is the Taylor series of $f$ at $a$ . Analyticity means that this series actually converges to the function in a neighborhood—not merely that its coefficients can be calculated. Inside its convergence disk, a power series converges uniformly on smaller closed disks and may be differentiated term by term. (complexanalysis.org)

Real analyticity and smoothness

A real-analytic function is locally represented by a convergent series in real variables. Every real-analytic function is smooth, meaning that derivatives of all orders exist and are continuous. The converse fails: smoothness alone does not ensure that the Taylor series reproduces the function. (cs.cornell.edu)

A standard example is

g(x)= \begin{cases} e^{-1/x},&x>0,\\ 0,&x\leq 0. \end{cases}

This function is infinitely differentiable, and $g^{(n)}(0)=0$ for every nonnegative integer $n$ . Its Taylor series at zero is therefore identically zero, although $g(x)>0$ for every positive $x$ . It is not analytic at zero. The example shows that even a convergent Taylor series need not equal the smooth function from which it was constructed. (cs.cornell.edu)

Complex analyticity and holomorphicity

A function on an open subset of $\mathbb C$ is holomorphic if the limit

f'(z)=\lim_{h\to0}\frac{f(z+h)-f(z)}{h}

exists at every point. Here $h$ approaches zero from arbitrary complex directions. Complex differentiability is consequently more restrictive than differentiability along the real axis. For instance, complex conjugation $f(z)=\overline z$ is smooth as a map of two real coordinates but is nowhere complex differentiable. (ocw.mit.edu)

The central equivalence is

\text{holomorphic on }U \quad\Longleftrightarrow\quad \text{analytic on }U.

No separate assumption of higher derivatives is needed: complex differentiability throughout an open neighborhood implies derivatives of every order and a local Taylor expansion. Differentiability at a single isolated point is insufficient. (math.ucla.edu)

Writing $f=u+iv$ and $z=x+iy$ , complex differentiability implies the Cauchy–Riemann equations

u_x=v_y,\qquad u_y=-v_x.

Conversely, if the first partial derivatives of $u$ and $v$ are continuous on an open set and satisfy these equations there, $f$ is holomorphic. Merely satisfying the equations at one point, without suitable differentiability hypotheses, does not establish analyticity. (ocw.mit.edu)

Integral representation and rigidity

The Cauchy integral formula explains much of the strength of complex analyticity. If $f$ is holomorphic on a neighborhood of a closed disk and $C$ is its positively oriented boundary, then for every interior point $z$ ,

f(z)=\frac{1}{2\pi i}\int_C \frac{f(\zeta)}{\zeta-z}\,d\zeta.

Thus values inside the disk are determined by values on its boundary. Differentiating the formula gives all higher derivatives; expanding its kernel as a geometric series produces the Taylor expansion. (ocw.mit.edu)

If $|f|\leq M$ on a circle of radius $R$ centered at $a$ , the associated estimates give

|f^{(n)}(a)|\leq \frac{n!M}{R^n}.

These bounds connect the size of a function with the growth of its derivatives. (ocw.mit.edu)

Another rigidity property is the identity theorem. Two holomorphic functions on the same connected open domain are identical if they agree on a set having an accumulation point inside that domain. A holomorphic function that is not identically zero therefore has isolated zeros. Its local behavior near a zero $a$ has the form

f(z)=(z-a)^m h(z),\qquad h(a)\ne0,

where the positive integer $m$ is the zero’s multiplicity. (math.ucla.edu)

Examples and convergence boundaries

Every polynomial and the exponential function are analytic throughout the complex plane; such functions are called entire functions. Sine and cosine are also entire. A quotient of polynomials is analytic wherever its denominator is nonzero. (ocw.mit.edu)

Analyticity throughout a domain does not mean that one Taylor series represents the function everywhere in that domain. For example,

\frac{1}{1+x^2} =\sum_{n=0}^{\infty}(-1)^n x^{2n}, \qquad |x|<1.

The function is real analytic on the whole real line, but this particular series has radius of convergence $1$ . Its complex extension has singularities at $z=i$ and $z=-i$ , both one unit from the center. (jirka.org)

In general, the Taylor series of a holomorphic function converges throughout every disk centered at the expansion point and contained in its domain. It may extend farther if the function admits a holomorphic extension. A nonremovable singularity encountered by such a disk prevents further enlargement of the convergence radius. (math.ucla.edu)

The analytic continuation of a function extends its local representation beyond an initially specified domain. When continuation exists, the identity theorem controls uniqueness on connected overlaps; existence is a separate question. (math.ucla.edu)

Several variables

For several real or complex variables, analyticity is defined using a convergent multivariable power series,

f(z)=\sum_{\alpha}c_\alpha(z-a)^\alpha,

where $\alpha=(\alpha_1,\ldots,\alpha_d)$ is a tuple of nonnegative integers and

(z-a)^\alpha=\prod_{j=1}^{d}(z_j-a_j)^{\alpha_j}.

Convergence is required in a neighborhood of the center. A holomorphic function of several complex variables is real analytic in its real and imaginary coordinates, but the converse is false: a real-analytic expression may also depend on the conjugate coordinates $\overline z_j$ . (jirka.org)

One-variable conclusions must not be transferred indiscriminately to higher dimensions. For example, the holomorphic function $f(z_1,z_2)=z_1$ vanishes along the entire set $z_1=0$ ; its zeros are not isolated. This illustrates why agreement on an arbitrary set with an accumulation point is no longer sufficient for a several-variable identity theorem. (jirka.org)

Applications and scope

Analytic functions provide a bridge between complex differentiation and partial differential equations. The real and imaginary parts of a holomorphic function are harmonic: each satisfies Laplace’s equation,

u_{xx}+u_{yy}=0,\qquad v_{xx}+v_{yy}=0.

This relationship allows complex functions to describe two-dimensional potential problems, including electrostatic potentials and idealized fluid flow. However, constructing a harmonic conjugate globally can depend on the topology of the domain; local existence does not automatically give a single-valued global representation. (ocw.mit.edu)

References

Power series and analytic functionsjirka.org
Notes for Math 520: Complex Analysismath.ucla.edu
Taylor Series Representationscomplexanalysis.org
Calculus and analysis — Numerical Methods for Data Sciencecs.cornell.edu
Tasty Bits of Several Complex Variablesjirka.org
04 S18 Topic 2: Analytic functionsocw.mit.edu
04 S18 Topic 4: Cauchy's integral formulaocw.mit.edu
04 S18 Topic 5: Introduction to harmonic functionsocw.mit.edu