mathematics

Hartogs's theorem on separate holomorphicity

Hartogs's theorem on separate holomorphicity 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 Hartogs's theorem on separate holomorphicity rather than just read about it. In short: In mathematics, Hartogs's theorem is a fundamental result of Friedrich Hartogs in the theory of several complex variables. Roughly speaking, it states that a 'separately analytic' function is continuous.

Key takeaways

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

Reference excerpt

In mathematics, Hartogs's theorem is a fundamental result of Friedrich Hartogs in the theory of several complex variables. Roughly speaking, it states that a 'separately analytic' function is continuous. More precisely, if F : C n → C {\displaystyle F:{\textbf {C}}^{n}\to {\textbf {C}}} is a function which is analytic in each variable zi, 1 ≤ i ≤ n, while the other variables are held constant, then F is a continuous function. A corollary is that the function F is then in fact an analytic function in the n-variable sense (i.e. that locally it has a Taylor expansion). Therefore, 'separate analyticity' and 'analyticity' are coincident notions, in the theory of several complex variables. Starting with the extra hypothesis that the function is continuous (or bounded), the theorem is much easier to prove and in this form is known as Osgood's lemma. There is no analogue of this theorem for real variables. If we assume that a function

f : R n → R {\displaystyle f\colon {\textbf {R}}^{n}\to {\textbf {R}}} is differentiable (or even analytic) in each variable separately, it is not true that f {\displaystyle f} will necessarily be continuous. A counterexample in two dimensions is given by

f ( x , y ) = x y x 2 + y 2 . {\displaystyle f(x,y)={\frac {xy}{x^{2}+y^{2}}}.}

If in addition we define f ( 0 , 0 ) = 0 {\displaystyle f(0,0)=0} , this function has well-defined partial derivatives in x {\displaystyle x} and y {\displaystyle y} at the origin, but it is not continuous at origin. (Indeed, the limits along the lines x = y {\displaystyle x=y} and x = − y {\displaystyle x=-y} are not equal, so there is no way to extend the definition of f {\displaystyle f} to include the origin and have the function be continuous there.)

References Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992. Fuks, Boris Abramovich (1963). Theory of Analytic Functions of Several Complex Variables. American Mathematical Society. ISBN 978-1-4704-4428-0. {{cite book}}: ISBN / Date incompatibility (help) Hörmander, Lars (1990) [1966], An Introduction to Complex Analysis in Several Variables (3rd ed.), North Holland, ISBN 978-1-493-30273-4

External links "Hartogs theorem", Encyclopedia of Mathematics, EMS Press, 2001 [1994] This article incorporates material from Hartogs's theorem on separate analyticity on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

Worked examples

Example 1 — a first encounter with Hartogs's theorem on separate holomorphicity

Start with the simplest possible case. Write down what Hartogs's theorem on separate holomorphicity 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 Hartogs's theorem on separate holomorphicity 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 Hartogs's theorem on separate holomorphicity 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 Hartogs's theorem on separate holomorphicity

In research: Hartogs's theorem on separate holomorphicity 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 Hartogs's theorem on separate holomorphicity 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: Hartogs's theorem on separate holomorphicity 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 Hartogs's theorem on separate holomorphicity 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 Hartogs's theorem on separate holomorphicity in 20 minutes

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

Frequently asked questions

What is Hartogs's theorem on separate holomorphicity in simple terms?

Why does Hartogs's theorem on separate holomorphicity 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 Hartogs's theorem on separate holomorphicity?

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 Hartogs's theorem on separate holomorphicity.