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, complex, function and/or theory:
“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)
“The uses of travel are occasional, and short; but the best fruit it finds, when it finds it, is conversation; and this is a main function of life.”
—Ralph Waldo Emerson (1803–1882)
“We commonly say that the rich man can speak the truth, can afford honesty, can afford independence of opinion and action;—and that is the theory of nobility. But it is the rich man in a true sense, that is to say, not the man of large income and large expenditure, but solely the man whose outlay is less than his income and is steadily kept so.”
—Ralph Waldo Emerson (1803–1882)