mathematics

Hadamard three-lines theorem

Hadamard three-lines 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 Hadamard three-lines theorem rather than just read about it. In short: In complex analysis, a branch of mathematics, the Hadamard three-line theorem is a result about the behaviour of holomorphic functions defined in regions bounded by parallel lines in the complex plane. The theorem is named after the French mathematician Jacques Hadamard.

Key takeaways

Hadamard three-lines 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 Hadamard three-lines theorem to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Hadamard three-lines theorem from memory before moving on to harder problems.

Reference excerpt

In complex analysis, a branch of mathematics, the Hadamard three-line theorem is a result about the behaviour of holomorphic functions defined in regions bounded by parallel lines in the complex plane. The theorem is named after the French mathematician Jacques Hadamard.

Statement

Applications The three-line theorem can be used to prove the Hadamard three-circle theorem for a bounded continuous function g ( z ) {\displaystyle g(z)} on an annulus { z : r ≤ | z | ≤ R } , {\displaystyle \{z:r\leq |z|\leq R\},} holomorphic in the interior. Indeed applying the theorem to

f ( z ) = g ( e z ) , {\displaystyle f(z)=g(e^{z}),}

shows that, if

m ( s ) = sup | z | = e s | g ( z ) | , {\displaystyle m(s)=\sup _{|z|=e^{s}}|g(z)|,}

then log m ( s ) {\displaystyle \log \,m(s)} is a convex function of s . {\displaystyle s.}

The three-line theorem also holds for functions with values in a Banach space and plays an important role in complex interpolation theory. It can be used to prove Hölder's inequality for measurable functions

∫ | g h | ≤ ( ∫ | g | p ) 1 p ⋅ ( ∫ | h | q ) 1 q , {\displaystyle \int |gh|\leq \left(\int |g|^{p}\right)^{1 \over p}\cdot \left(\int |h|^{q}\right)^{1 \over q},}

where 1 p + 1 q = 1 , {\displaystyle {1 \over p}+{1 \over q}=1,} by considering the function

f ( z ) = ∫ | g | p z | h | q ( 1 − z ) . {\displaystyle f(z)=\int |g|^{pz}|h|^{q(1-z)}.}

See also Riesz–Thorin theorem Phragmén–Lindelöf principle

References Hadamard, Jacques (1896), "Sur les fonctions entières" (PDF), Bull. Soc. Math. Fr., 24: 186–187 (the original announcement of the theorem) Reed, Michael; Simon, Barry (1975), Methods of modern mathematical physics, Volume 2: Fourier analysis, self-adjointness, Elsevier, pp. 33–34, ISBN 0-12-585002-6 Ullrich, David C. (2008), Complex made simple, Graduate Studies in Mathematics, vol. 97, American Mathematical Society, pp. 386–387, ISBN 978-0-8218-4479-3

Worked examples

Example 1 — a first encounter with Hadamard three-lines theorem

Start with the simplest possible case. Write down what Hadamard three-lines 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 Hadamard three-lines 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 Hadamard three-lines 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 Hadamard three-lines theorem

In research: Hadamard three-lines 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 Hadamard three-lines 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: Hadamard three-lines theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Convex analysis, Theorems in complex analysis, so understanding it makes those chapters shorter.
In everyday life: Look for Hadamard three-lines 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 Hadamard three-lines theorem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Hadamard three-lines 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 Hadamard three-lines theorem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Hadamard three-lines theorem in simple terms?

Why does Hadamard three-lines 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 Hadamard three-lines 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 Hadamard three-lines theorem.