The fastest way to become strong in complex analysis is to stop treating examples as decorations and start treating them as tools. A theorem in this subject usually carries a sharp geometric and analytic message, but that message only becomes durable when you can build functions that test the boundary of the theorem. The point of this article is simple: build examples on purpose.
Complex analysis is unusually friendly to example-building because the subject has a small set of structural pieces that combine well. Holomorphic functions, meromorphic functions, singularities, power series, Möbius maps, logarithms and branches, and contour integrals all interact in predictable ways. Once you know what each piece preserves and what it can break, you can produce examples for almost any question you want to ask.
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A practical recipe has to do more than list famous functions. It should tell you what to choose first, what to compose next, and how to verify that the example actually tests the claim you care about. That is what this guide does.
Start by deciding what feature the example must control
Before writing a formula, name the feature you are trying to control. In complex analysis, most examples are built to control one of these:
- zeros and their multiplicities,
- poles and their orders,
- residues,
- boundary size on a circle or line,
- image geometry under a holomorphic map,
- branch behavior,
- growth at infinity,
- or failure of holomorphicity at a specific point.
This step prevents random function collecting. If your target is a residue computation, start with local singular structure. If your target is a conformal mapping question, start with geometry of the domain and range. If your target is a counterexample \to a false converse, identify exactly which hypothesis you want to drop.
A good habit is to write a one-line specification first, such as:
- “I need a meromorphic function on $\mathbb{C}$ with simple poles at $1$ and $-1$, residues $2$ and $-3$, and no other poles.”
- “I need a holomorphic function on the unit disk with zeros at $a$ and $b$ but bounded by $1$ in modulus.”
- “I need a function that is harmonic as a real-valued function but not the real part of a globally defined holomorphic function on this domain.”
Once the specification is clear, the recipe becomes mechanical.
The core construction kit
The most productive examples come from a short list of building blocks.
Rational functions for singularity control
If you want exact control of poles, orders, and residues, begin with rational functions. Near a finite point $a$, the term
creates a pole of order $m$. For simple poles, the coefficient $c$ is the residue. Sums of such terms let you prescribe local meromorphic data quickly.
For instance, the specification
is handled immediately by
If you also want the function to vanish at $z=0$, add a holomorphic correction chosen to enforce the condition. Since the singular data are already correct, the correction can be a polynomial.
Rational functions are also the first place to test false statements about boundedness, removable singularities, and behavior at infinity.
Power series for local behavior
If your question is local, power series are the cleanest language. To create a holomorphic function near $0$ with a zero of exact order $k$, write
To control the first few derivatives, choose coefficients directly. This is useful when testing statements about local injectivity, critical points, and multiplicity.
Power series are also ideal for constructing examples that agree to high order at a point but differ globally. That helps when studying identity theorem hypotheses. Two functions can match many derivatives at one point and still fail to agree unless all coefficients match on a connected domain.
Exponential, logarithm, and trigonometric functions for periodicity and covering behavior
The map $e^z$ is the standard source of examples about non-injectivity, periodicity, and omitted values. It never vanishes, wraps horizontal strips onto annuli or punctured sectors under suitable restriction, and turns additive structure into multiplicative structure.
The logarithm is the standard source of branch issues. If you want a function that is locally holomorphic but globally obstructed by topology, expressions involving $\log z$ or $\sqrt{z}$ on punctured domains and slit domains are usually the right place \to \begin.
Trigonometric functions, via their exponential definitions, give examples with controlled zeros and poles in lattices and are useful for residue practice and periodic meromorphic behavior.
Möbius transformations for geometry and normalization
Möbius maps
are the most efficient tool for geometry. They map generalized circles to generalized circles and can send three chosen points to three chosen points. If a problem involves the disk, upper half-plane, or circles, normalize first with a Möbius map.
A large share of “clever” examples in complex analysis are only Möbius maps plus one nonlinear map like $z^2$, $e^z$, or a branch of $\log z$.
A reliable recipe for building examples
Here is a practical workflow that works in many settings.
Step A: Build the local data first
If the theorem or problem mentions singularities, zeros, or derivatives at points, encode those first. Use rational terms for poles and power series factors for zeros. Do not worry yet about global growth or image shape.
Example target: a meromorphic function with a double pole at $0$, a simple pole at $2$, residue $5$ at $2$, and a zero at $1$.
Start with
Then enforce $f(1)=0$ by subtracting the constant $f_0(1)$. The singular data stay unchanged because constants are entire.
This pattern appears constantly: build singular structure, then correct with a holomorphic term.
Step B: Normalize the domain geometry
If the problem lives on the unit disk, upper half-plane, annulus, or slit plane, move your domain \to a standard one if needed. Möbius maps and simple powers reduce geometry \to a familiar setting where standard examples live.
For example, if you need a bounded holomorphic function on a disk centered at $a$ of radius $r$ with a zero at $w$, map to the unit disk by
then use a disk automorphism on $\phi(w)$, and compose back. This avoids ad hoc formulas and makes the example structurally transparent.
Step C: Choose a growth mechanism
Ask what should happen near the boundary or at infinity.
- Polynomial and rational terms give algebraic growth.
- $e^z$ gives rapid directional growth and periodicity in the imaginary direction.
- $e^{-z^2}$ decays in some directions and blows up in others, useful for contour estimates.
- Bounded disk maps and Blaschke factors control modulus on $|z|<1$.
- Logarithms and roots create controlled branch behavior.
Many mistakes come from skipping this step. A function can satisfy all local constraints and still have the wrong global size behavior.
Step D: Verify the exact claim you want to test
Do not stop when the formula “looks \right.” Check the specific property line by line.
If the example is for a theorem hypothesis, verify each hypothesis explicitly.
If it is a counterexample, verify the omitted hypothesis fails and all retained hypotheses hold.
If it is for intuition, compute the image of a few curves or level sets to see the geometry.
This verification stage is where example-building becomes proof training rather than pattern matching.
Worked example 1: Bounded holomorphic function on the disk with prescribed zeros
Suppose you want a bounded holomorphic function on the unit disk with zeros at $a,b\in \mathbb{D}$, and you want $|f(z)|\le 1$.
The right building block is the Blaschke factor
which maps the unit disk to itself and vanishes at $a$. Then
is holomorphic on $\mathbb{D}$, bounded by $1$ in modulus, and has zeros at $a$ and $b$ with multiplicity one unless $a=b$.
Why this example matters:
it trains three habits at once.
- Use an automorphism that matches the geometry of the domain.
- Build multiplicity by multiplying factors.
- Use structure-preserving pieces instead of guessing a polynomial.
This is the practical style you want in complex analysis.
Worked example 2: A removable singularity hiding inside a formula
Students often misclassify singularities from appearance alone. Build an example that forces classification by series.
Take
At first glance, $z=0$ looks singular. But
so
which extends holomorphically \to $z=0$ with value $1$. The visible denominator is not enough to determine the singularity type.
This example is easy, but it teaches a high-value rule: singularity classification is local analytic data, not surface syntax.
Worked example 3: Branch sensitivity on a punctured domain
To build intuition for branches, compare the punctured plane $\mathbb{C}\setminus\{0\}$ with a slit plane such as $\mathbb{C}\setminus (-\infty,0]$.
On the slit plane, a holomorphic branch of $\log z$ exists, so
defines a holomorphic square root. On the punctured plane, no global holomorphic branch of $\log z$ exists, so no global holomorphic square root of $z$ exists there either.
This is a model example for topology-sensitive holomorphic behavior. The local formula is the same, but the global domain structure decides whether the object exists.
When building examples in this area, always specify the domain first. In complex analysis, the same expression can be valid, invalid, or multi-valued depending on the domain.
Worked example 4: A contour-friendly integrand by design
Suppose you want an example suited for residue computation on large circles. The practical design choice is to use rational functions whose numerator degree is at least two lower than the denominator degree, so the integral over a large circle is easy to estimate.
For example,
has simple poles at $0,2i,-2i$. The residues are easy to compute, and the decay $O(1/|z|^2)$ at infinity makes large-circle arguments clean. This type of example is ideal for training contour method decisions because it separates analytic structure from estimation noise.
The lesson is not the exact formula. It is the design principle:
choose algebraic decay first when the contour argument depends on vanishing boundary contributions.
How to build counterexamples without wandering off-topic
A counterexample in complex analysis should fail one thing and preserve as much else as possible. The most useful counterexamples are minimal.
If you want to show that “harmonic” does not imply “globally the real part of a holomorphic function” on every domain, do not use a pathological function. Use the standard angle function issue on a punctured domain. The obstruction comes from domain topology, not from bad regularity.
If you want to show that pointwise convergence of holomorphic functions is not enough for a holomorphic limit, choose a sequence that behaves well on sparse sets but lacks local uniform control. If the theorem needs local uniform convergence, your counterexample should spotlight that exact gap.
The rule is:
change one hypothesis at a time. Complex analysis rewards sharp examples.
A checklist for evaluating whether your example is good
A good example in complex analysis is not merely correct. It is reusable.
Ask these questions:
- Does the formula make the target feature visible?
- Can I explain why each factor is present?
- Does it generalize \to a family of examples?
- Does it test a theorem hypothesis sharply?
- Can I verify the key claim with standard tools from the same chapter?
If the answer is yes, you are building mathematical equipment, not just solving one exercise.
Closing perspective
Building examples in complex analysis is a form of disciplined design. You decide what must be controlled, choose structure-preserving building blocks, normalize the geometry, tune growth, and verify the exact target. That process trains the same instincts used in proofs: isolate the invariant, choose the right representation, and check hypotheses precisely.
The subject becomes far less mysterious when you see how many examples come from a small toolkit used well. A Möbius map, a power series factor, a rational singular part, and one exponential or logarithm often carry the entire construction. The real progress is not memorizing more formulas. It is learning to assemble them on purpose.
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