mathematics

Pluriharmonic function

Pluriharmonic function 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 Pluriharmonic function rather than just read about it. In short: In mathematics, precisely in the theory of functions of several complex variables, a pluriharmonic function is a real valued function which is locally the real part of a holomorphic function of several complex variables. Sometimes such a function is referred to as n-harmonic function, where n ≥ 2 is the dimension of the complex domain where the function is defined.

Key takeaways

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

Reference excerpt

In mathematics, precisely in the theory of functions of several complex variables, a pluriharmonic function is a real valued function which is locally the real part of a holomorphic function of several complex variables. Sometimes such a function is referred to as n-harmonic function, where n ≥ 2 is the dimension of the complex domain where the function is defined. However, in modern expositions of the theory of functions of several complex variables it is preferred to give an equivalent formulation of the concept, by defining pluriharmonic function a complex valued function whose restriction to every complex line is a harmonic function with respect to the real and imaginary part of the complex line parameter.

Formal definition Definition 1. Let G ⊆ Cn be a complex domain and f : G → R be a C2 (twice continuously differentiable) function. The function f is called pluriharmonic if, for every complex line

{ a + b z ∣ z ∈ C } ⊂ C n {\displaystyle \{a+bz\mid z\in \mathbb {C} \}\subset \mathbb {C} ^{n}}

formed by using every couple of complex tuples a, b ∈ Cn, the function

z ↦ f ( a + b z ) {\displaystyle z\mapsto f(a+bz)}

is a harmonic function on the set

{ z ∈ C ∣ a + b z ∈ G } ⊂ C . {\displaystyle \{z\in \mathbb {C} \mid a+bz\in G\}\subset \mathbb {C} .}

Definition 2. Let M be a complex manifold and f : M → R be a C2 function. The function f is called pluriharmonic if

d d c f = 0. {\displaystyle dd^{c}f=0.}

Basic properties Every pluriharmonic function is a harmonic function, but not the other way around. Further, it can be shown that for holomorphic functions of several complex variables the real (and the imaginary) parts are locally pluriharmonic functions. However a function being harmonic in each variable separately does not imply that it is pluriharmonic.

See also Plurisubharmonic function Wirtinger derivatives

Notes

Historical references Gunning, Robert C.; Rossi, Hugo (1965), Analytic Functions of Several Complex Variables, Prentice-Hall series in Modern Analysis, Englewood Cliffs, N.J.: Prentice-Hall, pp. xiv+317, ISBN 9780821869536, MR 0180696, Zbl 0141.08601. Krantz, Steven G. (1992), Function Theory of Several Complex Variables, Wadsworth & Brooks/Cole Mathematics Series (Second ed.), Pacific Grove, California: Wadsworth & Brooks/Cole, pp. xvi+557, ISBN 0-534-17088-9, MR 1162310, Zbl 0776.32001. Poincaré, H. (1899), "Sur les propriétés du potentiel et sur les fonctions Abéliennes", Acta Mathematica (in French), 22 (1): 89–178, doi:10.1007/BF02417872, JFM 29.0370.02. Severi, Francesco (1958), Lezioni sulle funzioni analitiche di più variabili complesse – Tenute nel 1956–57 all'Istituto Nazionale di Alta Matematica in Roma (in Italian), Padova: CEDAM – Casa Editrice Dott. Antonio Milani, pp. XIV+255, Zbl 0094.28002. Notes from a course held by Francesco Severi at the Istituto Nazionale di Alta Matematica (which at present bears his name), containing appendices of Enzo Martinelli, Giovanni Battista Rizza and Mario Benedicty. An English translation of the title reads as:-"Lectures on analytic functions of several complex variables – Lectured in 1956–57 at the Istituto Nazionale di Alta Matematica in Rome".

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Pluriharmonic function

Start with the simplest possible case. Write down what Pluriharmonic function 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 Pluriharmonic function 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 Pluriharmonic function 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 Pluriharmonic function

In research: Pluriharmonic function 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 Pluriharmonic function 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: Pluriharmonic function is common in secondary-school and first-year university syllabi. It links to neighbouring topics Harmonic functions, Several complex variables, so understanding it makes those chapters shorter.
In everyday life: Look for Pluriharmonic function 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 Pluriharmonic function in 20 minutes

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

Frequently asked questions

What is Pluriharmonic function in simple terms?

Why does Pluriharmonic function 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 Pluriharmonic function?

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 Pluriharmonic function.