In mathematics, precisely in the theory of functions of several complex variables, a pluriharmonic function is a real valued

function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-orie ...

which is

locally In mathematics, a mathematical object is said to satisfy a property locally, if the property is satisfied on some limited, immediate portions of the object (e.g., on some ''sufficiently small'' or ''arbitrarily small'' neighborhoods of points). P ...

the

real part In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...

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 In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...

of the

complex domain In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...

where the function is defined. However, in modern expositions of the theory of functions of several complex variablesSee for example the popular textbook by and the advanced (even if a little outdated) monograph by . 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 Line most often refers to: * Line (geometry), object with zero thickness and curvature that stretches to infinity * Telephone line, a single-user circuit on a telephone communication system Line, lines, The Line, or LINE may also refer to: Art ...

is a

harmonic function In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f: U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that is, ...

with respect to the

real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (201 ...

and

imaginary part In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...

of the complex line parameter.

Formal definition

. Let be a

and be a (twice

continuously differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...

) function. The function is called pluriharmonic if, for every

complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...

\ \subset \Complex^n

formed by using every couple of complex

tuple In mathematics, a tuple is a finite ordered list (sequence) of elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, referred to as ''the empty tuple''. An -tuple is defi ...

s , the function :

z \mapsto f(a + bz)

is a

on the set :

\ \subset \Complex .

. Let be a

complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic. The term complex manifold is variously used to mean a ...

and be a function. The function is called pluriharmonic if :

dd^c f = 0.

Basic properties

Every pluriharmonic function is a

, but not the other way around. Further, it can be shown that for

holomorphic function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex de ...

s 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.

Notes

Historical references

*. * . *. *. Notes from a course held by Francesco Severi at the

Istituto Nazionale di Alta Matematica The Istituto Nazionale di Alta Matematica Francesco Severi, abbreviated as INdAM, is a government created non-profit research institution whose main purpose is to promote research in the field of mathematics and its applications and the diffusion ...

(which at present bears his name), containing appendices of

Enzo Martinelli Enzo Martinelli (11 November 1911 – 27 August 1999 writes that his death year is 1998, unlike to , and , but it is probably a typographical error.) was an Italian mathematician, working in the theory of functions of several complex variables: ...

Giovanni Battista Rizza Giovanni Battista Rizza (7 February 1924 – 15 October 2018), officially known as Giambattista Rizza, was an Italian mathematician, working in the fields of complex analysis of several variables and in differential geometry: he is known for h ...

and

Mario Benedicty is a character created by Japanese video game designer Shigeru Miyamoto. He is the title character of the ''Mario'' franchise and the mascot of Japanese video game company Nintendo. Mario has appeared in over 200 video games since his crea ...

. 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''".

References

*. The first paper where a set of (fairly complicate) necessary and sufficient conditions for the solvability of the

Dirichlet problem In mathematics, a Dirichlet problem is the problem of finding a function which solves a specified partial differential equation (PDE) in the interior of a given region that takes prescribed values on the boundary of the region. The Dirichlet pro ...

for holomorphic functions of several variables is given. An English translation of the title reads as:-"''About a boundary value problem''". *."''Boundary value problems for pluriharmonic functions''" (English translation of the title) deals with

boundary value problem In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to ...

s for pluriharmonic functions: Fichera proves a trace condition for the solvability of the problem and reviews several earlier results of Enzo Martinelli, Giovanni Battista Rizza and Francesco Severi. *. An English translation of the title reads as:-"''Boundary values of pluriharmonic functions: extension to the space'' R^2''n'' ''of a theorem of L. Amoroso''". *. An English translation of the title reads as:-"''On a theorem of L. Amoroso in the theory of analytic functions of two complex variables''". *. *, available at Gallica *, available at Gallica *, available a
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External links