The Central Idea
A function
\[ f:\mathbb{C}\to\mathbb{C} \]is called analytic at a point \(z_0\) if it is complex differentiable at every point in some open neighborhood of \(z_0\).
Equivalently, there exists some radius \(r>0\) such that \(f\) is complex differentiable throughout the disk
\[ |z-z_0|Analytic and Holomorphic
In complex analysis, the terms analytic and holomorphic are commonly used for the same class of functions.
A function is holomorphic on an open set \(U\) if it is complex differentiable at every point of \(U\).
\[ \boxed{ f\text{ holomorphic on }U \iff f'(z)\text{ exists for every }z\in U. } \]Differentiable at a Point Versus Analytic
A function may possess a complex derivative at one isolated point without being analytic there.
For example,
\[ f(z)=z\overline{z} \]is complex differentiable at
\[ z=0, \]with
\[ f'(0)=0, \]but it is not differentiable throughout any disk around \(0\).
Therefore, it is not analytic at \(0\).
Why Open Sets Matter
Analyticity is naturally defined on open sets because the complex derivative requires nearby points in every direction.
A set \(U\subseteq\mathbb{C}\) is open if every point \(z_0\in U\) lies in some disk
\[ |z-z_0|Open sets provide enough surrounding space to study local complex behavior.
Analytic on a Domain
A domain is an open connected subset of the complex plane.
When a function is analytic throughout a domain \(D\), we may write
\[ \boxed{ f\in\mathcal{H}(D) } \]in notation often used for the family of holomorphic functions on \(D\).
Entire Functions
A function is called entire if it is analytic on all of \(\mathbb{C}\).
Examples include:
- polynomials;
- the complex exponential \(e^z\);
- \(\sin z\);
- \(\cos z\).
Thus,
\[ \boxed{ \text{entire} = \text{holomorphic everywhere in }\mathbb{C}. } \]Polynomials Are Entire
Let
\[ P(z)=a_nz^n+\cdots+a_1z+a_0. \]Every polynomial has a complex derivative at every point:
\[ P'(z) = na_nz^{n-1} +\cdots+a_1. \]Since there are no domain exclusions,
\[ \boxed{ P(z)\text{ is entire}. } \]Rational Functions Are Analytic on Their Domains
Consider
\[ f(z)=\frac{P(z)}{Q(z)}. \]Wherever
\[ Q(z)\neq0, \]the quotient rule applies and \(f\) is analytic.
Therefore,
\[ \boxed{ \frac{P(z)}{Q(z)} \text{ is analytic wherever }Q(z)\neq0. } \]Example: The Reciprocal Function
The function
\[ f(z)=\frac1z \]is not defined at \(z=0\), but it is analytic everywhere else.
Its derivative is
\[ \boxed{ f'(z)=-\frac1{z^2}, \qquad z\neq0. } \]Thus \(1/z\) is holomorphic on
\[ \boxed{ \mathbb{C}\setminus\{0\}. } \]The Exponential Function Is Entire
The complex exponential satisfies
\[ \frac{d}{dz}e^z=e^z. \]Its derivative exists at every complex number.
Therefore,
\[ \boxed{ e^z\text{ is entire}. } \]Complex Sine and Cosine Are Entire
Using the exponential definitions,
\[ \sin z = \frac{e^{iz}-e^{-iz}}{2i} \]and
\[ \cos z = \frac{e^{iz}+e^{-iz}}{2}, \]both functions are built from entire exponentials.
Therefore,
\[ \boxed{ \sin z\text{ and }\cos z\text{ are entire}. } \]Important Nonexamples
Several familiar complex-valued functions are not analytic.
Examples include
\[ \overline{z}, \qquad |z|, \qquad \operatorname{Re}(z), \qquad \operatorname{Im}(z). \]These functions are continuous, but they fail the directional consistency required for complex differentiability.
Why Analytic Functions Are Special
In real calculus, a function may possess one derivative without possessing many more.
Complex analysis is dramatically more rigid.
Once a function is analytic on an open set, deeper theorems imply that it is differentiable infinitely many times there.
Even more remarkably, it can be represented locally by a convergent power series.
Analytic Functions and Power Series
One of the central results of complex analysis is that a holomorphic function is locally representable by a power series.
Near a point \(z_0\),
\[ \boxed{ f(z) = \sum_{n=0}^{\infty} a_n(z-z_0)^n } \]within some disk of convergence.
The coefficients are determined by derivatives of the function.
The Complex Taylor Series
For an analytic function,
\[ a_n = \frac{f^{(n)}(z_0)}{n!}. \]Thus the local series has the form
\[ \boxed{ f(z) = \sum_{n=0}^{\infty} \frac{f^{(n)}(z_0)}{n!} (z-z_0)^n. } \]This is the complex Taylor expansion.
Power Series Produce Analytic Functions
Conversely, a convergent power series
\[ \sum_{n=0}^{\infty}a_n(z-z_0)^n \]defines an analytic function inside its disk of convergence.
The series may be differentiated term by term:
\[ \boxed{ f'(z) = \sum_{n=1}^{\infty} n a_n(z-z_0)^{n-1}. } \]Example: The Exponential Power Series
The complex exponential has the expansion
\[ \boxed{ e^z = \sum_{n=0}^{\infty} \frac{z^n}{n!}. } \]This series converges for every complex number \(z\).
Therefore, the series itself exhibits the entire nature of \(e^z\).
Complex Sine and Cosine Series
The familiar real power series remain valid for complex inputs:
\[ \boxed{ \sin z = \sum_{n=0}^{\infty} \frac{(-1)^n z^{2n+1}}{(2n+1)!} } \]and
\[ \boxed{ \cos z = \sum_{n=0}^{\infty} \frac{(-1)^n z^{2n}}{(2n)!}. } \]Both converge throughout the complex plane.
Local Information Determines More Than Expected
For analytic functions, derivatives at a point determine the entire local Taylor series.
This means that knowing
\[ f(z_0), \quad f'(z_0), \quad f''(z_0), \quad \ldots \]determines the function throughout a sufficiently small surrounding disk.
This rigidity is far stronger than what is typical for arbitrary real differentiable functions.
Zeros of Analytic Functions
Suppose an analytic function satisfies
\[ f(z_0)=0. \]If the function is not identically zero nearby, then there is a smallest positive integer \(m\) such that
\[ f^{(m)}(z_0)\neq0. \]Locally,
\[ f(z) = (z-z_0)^m g(z), \]where \(g\) is analytic and
\[ g(z_0)\neq0. \]The integer \(m\) is the multiplicity of the zero.
Simple Zeros
A zero \(z_0\) is called simple when
\[ f(z_0)=0 \]but
\[ f'(z_0)\neq0. \]In that case the multiplicity is \(1\).
Near \(z_0\), the function behaves approximately like
\[ f'(z_0)(z-z_0). \]Multiple Zeros
If
\[ f(z_0)=f'(z_0)=\cdots=f^{(m-1)}(z_0)=0 \]and
\[ f^{(m)}(z_0)\neq0, \]then \(z_0\) is a zero of multiplicity \(m\).
For example,
\[ f(z)=(z-2)^3 \]has a zero of multiplicity \(3\) at
\[ \boxed{z=2}. \]Zeros Are Usually Isolated
A nonzero analytic function cannot normally have zeros accumulating inside its domain.
Instead, its zeros occur as isolated points unless the function is identically zero.
This principle will later lead to the identity theorem.
A Preview of the Identity Theorem
Suppose two analytic functions agree on a set of points having an accumulation point inside a connected domain.
Then they must agree throughout the entire domain.
Symbolically, under the appropriate hypotheses,
\[ f(z)=g(z) \]on sufficiently many accumulating points implies
\[ \boxed{ f\equiv g. } \]This is another striking form of analytic rigidity.
Analyticity and the Component Functions
Write
\[ f(z)=u(x,y)+iv(x,y). \]Complex differentiability requires the real and imaginary components to satisfy special compatibility conditions.
The central equations are
\[ \boxed{u_x=v_y} \]and
\[ \boxed{u_y=-v_x}. \]These are the Cauchy–Riemann relationships.
Why the Cauchy–Riemann Relationships Matter
The equations connect horizontal change in one component to vertical change in the other.
They express the directional consistency required for the complex derivative to exist.
When the relevant partial derivatives are continuous, satisfying the Cauchy–Riemann equations throughout a region provides a practical route to establishing analyticity.
A Preview of Harmonic Functions
When an analytic function
\[ f=u+iv \]has sufficiently smooth component functions, both \(u\) and \(v\) satisfy Laplace's equation:
\[ \boxed{ u_{xx}+u_{yy}=0 } \]and
\[ \boxed{ v_{xx}+v_{yy}=0. } \]Such real-valued functions are called harmonic.
Why Analytic Components Become Harmonic
Start from
\[ u_x=v_y \]and
\[ u_y=-v_x. \]Differentiate the first equation with respect to \(x\):
\[ u_{xx}=v_{yx}. \]Differentiate the second with respect to \(y\):
\[ u_{yy}=-v_{xy}. \]If mixed partial derivatives agree,
\[ v_{yx}=v_{xy}, \]then
\[ \boxed{ u_{xx}+u_{yy}=0. } \]Harmonic Conjugates
If
\[ f=u+iv \]is analytic, then \(u\) and \(v\) are called harmonic conjugates of one another on the region, subject to the usual domain conditions.
Their gradients are related by a quarter-turn structure encoded in the Cauchy–Riemann equations.
Geometric Regularity of Analytic Functions
If
\[ f'(z_0)\neq0, \]then near \(z_0\),
\[ f(z)-f(z_0) \approx f'(z_0)(z-z_0). \]Multiplication by the nonzero complex number \(f'(z_0)\) produces local scaling and rotation.
This explains why analytic functions with nonzero derivative preserve local angles.
A Preview of Conformal Maps
A mapping is conformal at a point when it preserves angles and their orientation locally.
An analytic function with
\[ f'(z_0)\neq0 \]is conformal at \(z_0\).
Conformal mapping will later become a major application of analytic functions.
Operations That Preserve Analyticity
If \(f\) and \(g\) are analytic on a region, then the following are analytic wherever they are defined:
- \(f+g\);
- \(f-g\);
- \(fg\);
- \(cf\) for \(c\in\mathbb{C}\);
- \(f/g\), where \(g\neq0\);
- \(f\circ g\), when the composition is well-defined.
Composition of Analytic Functions
If \(g\) is analytic on a region and \(f\) is analytic on a region containing the image of \(g\), then
\[ f\circ g \]is analytic.
The derivative is given by
\[ \boxed{ (f\circ g)'(z) = f'(g(z))g'(z). } \]Example: An Entire Composition
Consider
\[ f(z)=e^{z^2}. \]The polynomial
\[ z^2 \]is entire, and \(e^z\) is entire.
Therefore,
\[ \boxed{ e^{z^2} \text{ is entire}. } \]The Complex Logarithm Requires a Domain Choice
The multivalued logarithm arises from
\[ z=re^{i\theta}. \]Formally,
\[ \log z = \ln r+i(\theta+2\pi k). \]To obtain a single-valued analytic logarithm, we must choose a continuous branch of the argument on a suitable domain.
This leads to the ideas of branches and branch cuts.
The Principal Logarithm
A common branch uses the principal argument
\[ -\pi<\operatorname{Arg}(z)<\pi. \]Then the principal logarithm is
\[ \boxed{ \operatorname{Log}z = \ln|z| + i\operatorname{Arg}(z). } \]This branch is analytic on the complex plane with the nonpositive real axis removed.
Square Roots and Branches
The equation
\[ w^2=z \]has two roots for \(z\neq0\).
A single-valued analytic square-root function therefore requires a domain on which one branch can be selected consistently.
As with logarithms, global single-valued choices can be obstructed by winding around the origin.
Analyticity Is Local
A function can be analytic on one region and fail to be analytic on another.
For example,
\[ \frac1z \]is analytic on every open set that avoids \(0\).
Analyticity is therefore always discussed relative to the region being considered.
Where Analyticity Fails
Points where a function fails to be analytic often carry important information.
For example,
\[ \frac1z \]is analytic everywhere except at \(z=0\).
Later, such exceptional points will be classified as removable singularities, poles, or essential singularities.
Analytic Functions Are Continuous
Complex differentiability implies continuity at each point.
Therefore, if \(f\) is analytic on a region \(D\), then
\[ \boxed{ f\text{ is continuous throughout }D. } \]Continuity is thus a necessary consequence of analyticity.
Continuity Is Far Weaker Than Analyticity
The function
\[ f(z)=\overline{z} \]is continuous everywhere, but analytic nowhere.
Therefore,
\[ \boxed{ \text{analytic} \Rightarrow \text{continuous}, } \]but not conversely.
Analyticity Is Stronger Than Ordinary Smoothness
A function
\[ f(z)=u(x,y)+iv(x,y) \]may have component functions possessing many ordinary partial derivatives and still fail to be analytic.
The issue is not merely smoothness.
The real and imaginary parts must change in a coordinated way compatible with multiplication by complex numbers.
Example: A Local Power Series for a Rational Function
Consider
\[ f(z)=\frac1{1-z}. \]For
\[ |z|<1, \]the geometric-series formula gives
\[ \boxed{ \frac1{1-z} = 1+z+z^2+z^3+\cdots. } \]This is a local analytic expansion centered at \(0\).
What Limits a Taylor Series?
A Taylor series centered at \(z_0\) converges at least until it encounters an obstruction to analyticity.
For
\[ f(z)=\frac1{1-z}, \]the nearest singularity to \(0\) is
\[ z=1. \]Its distance from the center is \(1\), matching the radius of convergence of the geometric series.
Common Error: Confusing Pointwise Differentiability with Analyticity
Showing that \(f'(z_0)\) exists at one point does not prove that \(f\) is analytic at \(z_0\).
Analyticity requires differentiability throughout some neighborhood of the point.
Common Error: Calling Every Analytic Function Entire
An analytic function may have a restricted domain.
The function
\[ \frac1z \]is analytic on
\[ \mathbb{C}\setminus\{0\}, \]but it is not entire because it is not analytic at \(0\).
Common Error: Using the Cauchy–Riemann Equations Without Checking Regularity
The Cauchy–Riemann relationships are central, but their use as a sufficient test requires appropriate assumptions about the component functions.
Do not treat the equations as an automatic substitute for all other differentiability conditions.
Common Error: Confusing Local and Global Properties
Analyticity is local.
A function may be analytic in many small neighborhoods while global properties such as invertibility, logarithm branches, or square-root branches still depend on the shape of the entire domain.
Common Error: Assuming “Analytic” Means the Same Thing in Every Context
In real analysis, a function may be infinitely differentiable without being represented by its Taylor series.
In complex analysis, holomorphic functions are automatically locally representable by convergent power series.
This stronger equivalence is one of the defining features of the subject.
A Reliable Procedure for Testing Analyticity
- Identify the domain and any excluded points.
- Determine whether the function is complex differentiable throughout an open neighborhood.
- When useful, write \(f=u+iv\).
- Compute the relevant partial derivatives.
- Check the Cauchy–Riemann relationships.
- Verify the regularity assumptions required by the theorem being used.
- State clearly the region on which the function is analytic.
A Reliable Procedure for Building Analytic Functions
- Start from known analytic functions such as polynomials and \(e^z\).
- Form sums, differences, and products.
- Use quotients away from zeros of the denominator.
- Compose analytic functions when domains are compatible.
- Track all domain restrictions.
- Identify singular points separately.
Worked Example: Determine Where a Rational Function Is Analytic
Consider
\[ f(z)=\frac{z^2+1}{z^2-4}. \]Factor the denominator:
\[ z^2-4=(z-2)(z+2). \]Therefore, the excluded points are
\[ z=2 \]and
\[ z=-2. \]Since rational functions are analytic wherever the denominator is nonzero,
\[ \boxed{ f\text{ is analytic on } \mathbb{C}\setminus\{-2,2\}. } \]Worked Example: An Entire Function
Consider
\[ f(z)=z^3e^z+\sin z. \]The functions
\[ z^3, \qquad e^z, \qquad \sin z \]are entire.
Products and sums of entire functions remain entire.
Therefore,
\[ \boxed{ z^3e^z+\sin z \text{ is entire}. } \]Worked Example: A Function That Is Not Analytic
Consider
\[ f(z)=\overline{z}. \]The previous lesson showed that the complex derivative fails at every point because the difference quotient depends on the direction of approach.
Therefore,
\[ \boxed{ f(z)=\overline{z} \text{ is analytic nowhere}. } \]Worked Example: A Local Series
Expand
\[ \frac1{1-z} \]around \(z=0\).
Using the geometric series,
\[ \frac1{1-z} = \sum_{n=0}^{\infty}z^n \]for
\[ |z|<1. \]Therefore,
\[ \boxed{ \frac1{1-z} = 1+z+z^2+z^3+\cdots, \qquad |z|<1. } \]Practice 1: Entire or Not?
Determine whether
\[ f(z)=4z^5-3z+7 \]is entire.
Practice 2: Rational Analyticity
Determine where
\[ f(z)=\frac{z+1}{z^2+1} \]is analytic.
Practice 3: Composition
Determine whether
\[ f(z)=e^{z^3} \]is entire.
Practice 4: Nonexample
Is
\[ f(z)=|z| \]analytic anywhere?
Practice 5: Reciprocal Function
State the largest natural domain on which
\[ f(z)=\frac1{z-3} \]is analytic.
Practice 6: Entire Combination
Determine whether
\[ f(z)=z^2\cos z+e^z \]is entire.
Practice 7: Zero Multiplicity
Find the multiplicity of the zero at \(z=1\) for
\[ f(z)=(z-1)^4(z+2). \]Practice 8: Simple Zero
Determine whether \(z=0\) is a simple zero of
\[ f(z)=\sin z. \]Practice 9: Local Power Series
Write the first four terms of the expansion
\[ \frac1{1-z} \]about \(z=0\).
Practice 10: Analyticity and Continuity
If a function is analytic at \(z_0\), what can you conclude immediately about continuity at \(z_0\)?
Practice Answers
Practice 1
The function is a polynomial, so it is complex differentiable everywhere.
\[ \boxed{ f\text{ is entire}. } \]Practice 2
The denominator vanishes when
\[ z^2+1=0. \]Thus
\[ z=\pm i. \]Therefore,
\[ \boxed{ f\text{ is analytic on } \mathbb{C}\setminus\{-i,i\}. } \]Practice 3
The polynomial \(z^3\) is entire and \(e^z\) is entire.
Their composition is entire.
\[ \boxed{ e^{z^3}\text{ is entire}. } \]Practice 4
The modulus function is continuous everywhere but complex differentiable nowhere.
\[ \boxed{ |z|\text{ is analytic nowhere}. } \]Practice 5
The only excluded point is
\[ z=3. \]Therefore,
\[ \boxed{ \mathbb{C}\setminus\{3\}. } \]Practice 6
The functions \(z^2\), \(\cos z\), and \(e^z\) are entire.
Products and sums preserve entire-ness.
\[ \boxed{ z^2\cos z+e^z \text{ is entire}. } \]Practice 7
The factor
\[ (z-1)^4 \]shows that the zero at \(z=1\) has multiplicity
\[ \boxed{4}. \]Practice 8
We have
\[ \sin0=0 \]and
\[ (\sin z)'_{z=0} = \cos0 = 1\neq0. \]Therefore,
\[ \boxed{ z=0\text{ is a simple zero of }\sin z. } \]Practice 9
\[ \frac1{1-z} = 1+z+z^2+z^3+\cdots. \]The first four terms are
\[ \boxed{ 1+z+z^2+z^3. } \]Practice 10
Analyticity implies complex differentiability in a neighborhood, and complex differentiability implies continuity.
Therefore,
\[ \boxed{ f\text{ is continuous at }z_0. } \]Checking Strategies
- Always state the region on which analyticity is being claimed.
- For rational functions, locate zeros of the denominator first.
- Use closure properties to recognize analytic combinations efficiently.
- Distinguish analyticity at a point from differentiability at one isolated point.
- For functions written as \(u+iv\), use component derivatives and the Cauchy–Riemann relationships carefully.
- For Taylor expansions, identify the nearest obstruction to analyticity when reasoning about radius of convergence.
Connections to Earlier Mathematics
Analytic functions unify several major ideas:
- Complex differentiation: analyticity requires differentiability throughout open neighborhoods.
- Power series: analytic functions admit local Taylor expansions.
- Multivariable calculus: real and imaginary components satisfy coupled partial-derivative conditions.
- Differential equations: harmonic component functions satisfy Laplace's equation.
- Geometry: nonzero derivatives create local conformal behavior.
What to Remember
A function is analytic at \(z_0\) when it is complex differentiable throughout some neighborhood of \(z_0\).
\[ \boxed{ \text{analytic} = \text{holomorphic}. } \]A function analytic on all of \(\mathbb{C}\) is
\[ \boxed{\text{entire}}. \]Analytic functions are continuous, infinitely differentiable, and locally representable by convergent power series.
For
\[ f=u+iv, \]the component functions satisfy the key relationships
\[ \boxed{u_x=v_y} \]and
\[ \boxed{u_y=-v_x}. \]These equations provide the next major structural step in the theory.
Where We Go Next
Analytic functions possess a remarkable amount of structure, but we still need a practical way to detect whether a function
\[ f(z)=u(x,y)+iv(x,y) \]can be complex differentiable.
The next lesson, Cauchy–Riemann Relationships, derives the equations \[ u_x=v_y, \qquad u_y=-v_x, \] explains their geometric meaning, and develops them into a working test for complex differentiability and analyticity.