Harmonic Function - Connections With Complex Function Theory

Connections With Complex Function Theory

The real and imaginary part of any holomorphic function yield harmonic functions on R2 (these are said to be a pair of harmonic conjugate functions). Conversely, any harmonic function u on an open subset Ω of R2 is locally the real part of a holomorphic function. This is immediately seen observing that, writing z = x + iy, the complex function g(z) := ux − i uy is holomorphic in Ω because it satisfies the Cauchy–Riemann equations. Therefore, g has locally a primitive f, and u is the real part of f up to a constant, as ux is the real part of .

Although the above correspondence with holomorphic functions only holds for functions of two real variables, still harmonic functions in n variables enjoy a number of properties typical of holomorphic functions. They are (real) analytic; they have a maximum principle and a mean-value principle; a theorem of removal of singularities as well as a Liouville theorem one holds for them in analogy to the corresponding theorems in complex functions theory.

Read more about this topic:  Harmonic Function

Famous quotes containing the words connections with, connections, complex, function and/or theory:

    Growing up human is uniquely a matter of social relations rather than biology. What we learn from connections within the family takes the place of instincts that program the behavior of animals; which raises the question, how good are these connections?
    Elizabeth Janeway (b. 1913)

    Imagination is an almost divine faculty which, without recourse to any philosophical method, immediately perceives everything: the secret and intimate connections between things, correspondences and analogies.
    Charles Baudelaire (1821–1867)

    Instead of seeing society as a collection of clearly defined “interest groups,” society must be reconceptualized as a complex network of groups of interacting individuals whose membership and communication patterns are seldom confined to one such group alone.
    Diana Crane (b. 1933)

    Morality and its victim, the mother—what a terrible picture! Is there indeed anything more terrible, more criminal, than our glorified sacred function of motherhood?
    Emma Goldman (1869–1940)

    Everything to which we concede existence is a posit from the standpoint of a description of the theory-building process, and simultaneously real from the standpoint of the theory that is being built. Nor let us look down on the standpoint of the theory as make-believe; for we can never do better than occupy the standpoint of some theory or other, the best we can muster at the time.
    Willard Van Orman Quine (b. 1908)