Nachbin's theorem

Key takeaways

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

Reference excerpt

In mathematics, in the area of complex analysis, Nachbin's theorem (named after Leopoldo Nachbin) is a result used to establish bounds on the growth rates for analytic functions. In particular, Nachbin's theorem may be used to give the domain of convergence of the generalized Borel transform, also called Nachbin summation. This article provides a brief review of growth rates, including the idea of a function of exponential type. Classification of growth rates based on type help provide a finer tool than big O or Landau notation, since a number of theorems about the analytic structure of the bounded function and its integral transforms can be stated.

Exponential type

A function f ( z ) {\displaystyle f(z)} defined on the complex plane is said to be of exponential type if there exist constants M {\displaystyle M} and α {\displaystyle \alpha } such that

| f ( r e i θ ) | ≤ M e α r {\displaystyle |f(re^{i\theta })|\leq Me^{\alpha r}}

in the limit of r → ∞ {\displaystyle r\to \infty } . Here, the complex variable z {\displaystyle z} was written as z = r e i θ {\displaystyle z=re^{i\theta }} to emphasize that the limit must hold in all directions θ {\displaystyle \theta } . Letting α {\displaystyle \alpha } stand for the infimum of all such α {\displaystyle \alpha } , one then says that the function f {\displaystyle f} is of exponential type α {\displaystyle \alpha } . For example, let f ( z ) = sin ⁡ ( π z ) {\displaystyle f(z)=\sin(\pi z)} . Then one says that sin ⁡ ( π z ) {\displaystyle \sin(\pi z)} is of exponential type π {\displaystyle \pi } , since π {\displaystyle \pi } is the smallest number that bounds the growth of sin ⁡ ( π z ) {\displaystyle \sin(\pi z)} along the imaginary axis. So, for this example, Carlson's theorem cannot apply, as it requires functions of exponential type less than π {\displaystyle \pi } .

Ψ type Additional function types may be defined for other bounding functions besides the exponential function. In general, a function Ψ ( t ) {\displaystyle \Psi (t)} is a comparison function if it has a series

Ψ ( t ) = ∑ n = 0 ∞ Ψ n t n {\displaystyle \Psi (t)=\sum _{n=0}^{\infty }\Psi _{n}t^{n}}

with Ψ n > 0 {\displaystyle \Psi _{n}>0} for all n {\displaystyle n} , and

lim n → ∞ Ψ n + 1 Ψ n = 0. {\displaystyle \lim _{n\to \infty }{\frac {\Psi _{n+1}}{\Psi _{n}}}=0.}

Comparison functions are necessarily entire, which follows from the ratio test. If Ψ ( t ) {\displaystyle \Psi (t)} is such a comparison function, one then says that f {\displaystyle f} is of Ψ {\displaystyle \Psi } -type if there exist constants M {\displaystyle M} and τ {\displaystyle \tau } such that

| f ( r e i θ ) | ≤ M Ψ ( τ r ) {\displaystyle \left|f\left(re^{i\theta }\right)\right|\leq M\Psi (\tau r)}

as r → ∞ {\displaystyle r\to \infty } . If τ {\displaystyle \tau } is the infimum of all such τ {\displaystyle \tau } one says that f {\displaystyle f} is of Ψ {\displaystyle \Psi } -type τ {\displaystyle \tau } . Nachbin's theorem states that a function f ( z ) {\displaystyle f(z)} with the series

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Nachbin's theorem

Start with the simplest possible case. Write down what Nachbin'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 Nachbin'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 Nachbin'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 Nachbin's theorem

In research: Nachbin'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 Nachbin'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: Nachbin's theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Integral transforms, Summability methods, Theorems in complex analysis, so understanding it makes those chapters shorter.
In everyday life: Look for Nachbin'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.

How to study Nachbin's theorem in 20 minutes

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

Frequently asked questions

What is Nachbin's theorem in simple terms?

In mathematics, in the area of complex analysis, Nachbin's theorem (named after Leopoldo Nachbin) is a result used to establish bounds on the growth rates for analytic functions. In particular, Nachbin's theorem may be used to give the domain of convergence of the generalized Borel transform, also…

Why does Nachbin'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 Nachbin'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 Nachbin's theorem.