# Harmonic Functions

The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

## Harmonic Functions

functions of n variables (n≤2) that are continuous over some domain together with their partial derivatives of the first and second orders and that satisfy Laplace’s differential equation in this domain: In many problems of physics and mechanics concerned with the state of a portion of space, which depends on the position of a point but not on time (equilibrium, steady-state motion, and so on), the corresponding state is represented by a harmonic function of the point’s coordinates. For example, the potential of the gravitational forces in a region that contains no attracting masses and the potential of a constant electrical field in a region containing no electrical charges are harmonic functions. The potential of the velocities of a steady-state irrotational motion of an incompressible fluid, the temperature of a body under conditions of a steady-state heat distribution, and the bending of a membrane stretched over an arbitrary contour that is completely nonplanar (the membrane’s weight is ignored) are also harmonic functions.

Harmonic functions of three variables (the coordinates of a point) are the most important for use in physics and mechanics. In a particular case, when the region of space is bounded by a cylindrical surface whose generatrices are parallel—for example, the z axes—the phenomenon under investigation proceeds in the same manner in any plane perpendicular to the generatrices (it does not depend on the z coordinate), the corresponding harmonic functions of three variables are converted into harmonic functions of two variables, x and y. The latter are closely related to the analytic functions f(ξ) of a complex variable ξ = x + iy. Each harmonic function of x and y is a real or imaginary part of some function f(ξ), and conversely, the real and imaginary parts of any analytic function are harmonic functions of x and y. For example, x2y2 and 2 xy, being the real and imaginary parts of the function ξ2 = x2y2 + 2 ixy, are harmonic functions. The major problems of the theory of harmonic functions are boundary-value problems, in which it is necessary to find a harmonic function within a region using data concerning the behavior of the function on the boundary of the region. An example is the Dirichlet problem, in which a harmonic function is sought from its values on the region’s boundary points (for example, the determination of the temperature within a body from its surface temperature, which is maintained in such a way that it depends on position but not on time, or the determination of the form of a membrane from the form of the contour over which it is stretched). There is also the Neumann problem, in which a harmonic function is sought from the value of its normal derivative, which is specified on the boundary of a region (for example, the determination of the temperature within a body from the temperature gradient on its surface, or the determination of the potential of the motion of an incompressible fluid around a solid object, based on the fact that the normal components of the velocities of the particles of fluid that adjoin the object’s surface coincide with the specified normal components of the velocities of points on the surface).

Various methods of great theoretical value have been developed to solve the Dirichlet, Neumann, and other boundary-value problems of the theory of harmonic functions. Known methods include, for the Dirichlet problem, the alternating method (Schwartz’s method), the sweeping-out method (Poincaré’s method), the method of integral equations (Fredholm’s method), and the method of upper and lower functions (Perron’s method). In considering boundary-value problems for regions of a general form, important problems arise concerning the conditions of existence of solutions and the stability of solutions for small variations of the region’s boundary. These problems are the topic of the work of M. V. Keldysh, M. A. Lavrent’ev, and other contemporary Soviet mathematicians. Numerical methods of solving boundary-value problems have been developed for use in physics and technology.

### REFERENCES

Keldysh, M. V. “O razreshimosti i ustoichivosti zadachi Dirikhle.” Uspekhi matematicheskikh nauk, 1940, issue 8.
Sretenskii, L. N. Teoriia n’iutonovskogo potentsiala. Moscow-Leningrad, 1946.
Smirnov, V. I. Kurs vysshei matematiki, 3rd ed., vol. 4. Moscow, 1957.
Petrovskii, I. G. Lektsii ob uravneniiakh s chastnymi proizvodnymi, 3rd ed. Moscow, 1961.

A. I. MARKUSHEVICH

References in periodicals archive ?
Kasue investigated the theory of harmonic functions for such manifolds.
Consider the family of complex-valued harmonic functions f = u + iv defined in D, where u and v are real harmonic in D.
Another group of normal and shear stresses is produced again when the Boussinesq solutions are applied to correct stresses on the sloping surface, although the original stresses on horizontal free surface have been removed by introduction of harmonic functions (see (39) and (48)).
By applying an affine transformation ([bar.[alpha]]f - [[bar.[beta]]f] - [bar.[alpha]] [a.sub.0] + [bar.[beta]][a.sub.0]/[([absolute value of [alpha]].sup.2] - [[absolute value of [beta]].sup.2], we may restrict our attention to the family [[summation].sub.H] of harmonic functions of the form
It provides an equivalence theorem proving that the definition of Hardy spaces by conjugate harmonic functions is equivalent to the atomic definition of these spaces.
Mathematicians from Europe and the US discuss geometry, algebraic geometry, and topology, including the basic properties of uniformly rational varieties, new relations between algebraic topology and the theory of Hopf, a measured foliated 2-complex thin type, the classical geometry of complexes, a generalization of the amoeba and the Ronkin function of a plane algebraic curve for a pair of harmonic functions, and natural differential-geometric constructions on the algebra of densities.
This means that f is a map from G into [R.sup.2] and both [f.sub.1] and [f.sub.2] are harmonic functions, that is, solutions of the two-dimensional Laplace equation
where G and K are complex-valued harmonic functions in D (cf.
The first five chapters form the core material of the text, covering fundamental concepts, holomorphic and harmonic functions, Cauchy theory and its applications, and isolated singularities.
Subsequent chapters discuss field equations of flow through a porous medium discharge potentials for two-dimensional flows in horizontal, shallow aquifers, the Laplace equation, harmonic functions, and method of images, flow net, determination of aquifer characteristics, and the finite element method.
Coverage includes complex numbers, the concept of homotopy and its application, integral theorems, the theory and application of harmonic functions, and the correspondence of boundaries under conformal mapping.
The topics include Brownian motion and harmonic measure in conic sections, the sharpness of certain approach regions, integral representation for space-time excessive functions, hyperbolic Riemann surfaces without unbounded positive harmonic functions, and a vanishing theorem on the point- wise defect of a rational iteration sequence for moving targets.

Site: Follow: Share:
Open / Close