mathematics

Lindelöf's theorem

Lindelöf's theorem is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Lindelöf's theorem rather than just read about it. In short: In mathematics, Lindelöf's theorem is a result in complex analysis named after the Finnish mathematician Ernst Leonard Lindelöf. It states that a holomorphic function on a half-strip in the complex plane that is bounded on the boundary of the strip and does not grow "too fast" in the unbounded direction of the strip must remain bounded on the whole strip.

Key takeaways

Lindelöf's theorem belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Lindelöf's theorem to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Lindelöf's theorem from memory before moving on to harder problems.

Reference excerpt

In mathematics, Lindelöf's theorem is a result in complex analysis named after the Finnish mathematician Ernst Leonard Lindelöf. It states that a holomorphic function on a half-strip in the complex plane that is bounded on the boundary of the strip and does not grow "too fast" in the unbounded direction of the strip must remain bounded on the whole strip. The result is useful in the study of the Riemann zeta function, and is a special case of the Phragmén–Lindelöf principle. Also, see Hadamard three-lines theorem.

Statement of the theorem Let Ω {\displaystyle \Omega } be a half-strip in the complex plane:

Ω = { z ∈ C | x 1 ≤ R e ( z ) ≤ x 2 and I m ( z ) ≥ y 0 } ⊊ C . {\displaystyle \Omega =\{z\in \mathbb {C} |x_{1}\leq \mathrm {Re} (z)\leq x_{2}\ {\text{and}}\ \mathrm {Im} (z)\geq y_{0}\}\subsetneq \mathbb {C} .}

Suppose that f {\displaystyle f} is holomorphic (i.e. analytic) on Ω {\displaystyle \Omega } and that there are constants

M {\displaystyle M} , A {\displaystyle A} , and B {\displaystyle B} such that

| f ( z ) | ≤ M for all z ∈ ∂ Ω {\displaystyle |f(z)|\leq M\ {\text{for all}}\ z\in \partial \Omega }

and

| f ( x + i y ) | ≤ B y A for all x + i y ∈ Ω . {\displaystyle |f(x+iy)|\leq By^{A}\ {\text{for all}}\ x+iy\in \Omega .}

Then f {\displaystyle f} is bounded by M {\displaystyle M} on all of Ω {\displaystyle \Omega } :

| f ( z ) | ≤ M for all z ∈ Ω . {\displaystyle |f(z)|\leq M\ {\text{for all}}\ z\in \Omega .}

Proof Fix a point ξ = σ + i τ {\displaystyle \xi =\sigma +i\tau } inside Ω {\displaystyle \Omega } . Choose λ > − y 0 {\displaystyle \lambda >-y_{0}} , an integer N > A {\displaystyle N>A} and y 1 > τ {\displaystyle y_{1}>\tau } large enough such that

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Lindelöf's theorem

Start with the simplest possible case. Write down what Lindelöf's theorem claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Lindelöf's theorem before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Lindelöf's theorem ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Lindelöf's theorem

In research: Lindelöf's theorem appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Lindelöf's theorem in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Lindelöf's theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Theorems in complex analysis, so understanding it makes those chapters shorter.
In everyday life: Look for Lindelöf's theorem outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Lindelöf's theorem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Lindelöf's theorem means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Lindelöf's theorem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Lindelöf's theorem in simple terms?

Why does Lindelöf's theorem matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Lindelöf's theorem?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Lindelöf's theorem.