Hefer's theorem

Key takeaways

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

Reference excerpt

In several complex variables, Hefer's theorem is a result that represents the difference at two points of a holomorphic function as the sum of the products of the coordinate differences of these two points with other holomorphic functions defined in the Cartesian product of the function's domain. The theorem bears the name of Hans Hefer. The result was published by Karl Stein and Heinrich Behnke under the name Hans Hefer. In a footnote in the same article, it is written that Hans Hefer died on the eastern front and that the work was an excerpt from Hefer's dissertation which he defended in 1940.

Statement of the theorem Let Ω ⊂ C n {\displaystyle \Omega \subset \mathbb {C} ^{n}} be a domain of holomorphy and f : Ω ↦ C {\displaystyle f:\Omega \mapsto \mathbb {C} } be a holomorphic function. Then, there exist holomorphic functions g 1 , ⋯ , g n {\displaystyle g_{1},\cdots ,g_{n}} defined on Ω × Ω {\displaystyle \Omega \times \Omega } so that

f ( z ) − f ( w ) = ∑ j = 1 n ( z j − w j ) g j ( w , z ) {\displaystyle f(z)-f(w)=\sum _{j=1}^{n}(z_{j}-w_{j})g_{j}(w,z)}

holds for every z , w ∈ Ω {\displaystyle z,w\in \Omega } . The decomposition in the theorem is feasible also on many non-pseudoconvex domains.

Hefer's lemma The proof of the theorem follows from Hefer's lemma. Let Ω ⊂ C n {\displaystyle \Omega \subset \mathbb {C} ^{n}} be a domain of holomorphy and f : Ω ↦ C {\displaystyle f:\Omega \mapsto \mathbb {C} } be a holomorphic function. Suppose that f {\displaystyle f} is identically zero on the intersection of Ω {\displaystyle \Omega } with the ( N − k ) {\displaystyle (N-k)} -dimensional complex coordinate space; i.e.

f ( 0 , ⋯ , 0 , z k + 1 , z k , ⋯ , z n ) ≡ 0 {\displaystyle f(0,\cdots ,0,z_{k+1},z_{k},\cdots ,z_{n})\equiv 0} . Then, there exist holomorphic functions g 1 , ⋯ , g n {\displaystyle g_{1},\cdots ,g_{n}} defined on Ω {\displaystyle \Omega } so that

f ( z ) = ∑ j = 1 n z j g j ( z ) {\displaystyle f(z)=\sum _{j=1}^{n}z_{j}g_{j}(z)}

holds for every z ∈ Ω {\displaystyle z\in \Omega } .

References

Worked examples

Example 1 — a first encounter with Hefer's theorem

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

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

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

Frequently asked questions

What is Hefer's theorem in simple terms?

In several complex variables, Hefer's theorem is a result that represents the difference at two points of a holomorphic function as the sum of the products of the coordinate differences of these two points with other holomorphic functions defined in the Cartesian product of the function's domain. T…

Why does Hefer'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 Hefer'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 Hefer's theorem.