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Introduction to complex analysis
Introduction to complex analysis

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2.4 Laplace’s equation and electrostatics

The Cauchy–Riemann equations for a differentiable function f times left parenthesis x plus i times y right parenthesis equals u of x comma y plus i times v of x comma y tell us that

prefix partial differential of of u divided by prefix partial differential of of x times left parenthesis x comma y right parenthesis equals prefix partial differential of of v divided by prefix partial differential of of y times left parenthesis x comma y right parenthesis and prefix partial differential of of v divided by prefix partial differential of of x times left parenthesis x comma y right parenthesis equals negative prefix partial differential of of u divided by prefix partial differential of of y times left parenthesis x comma y right parenthesis full stop

These partial derivatives are themselves functions of x and y, so, provided that they are suitably well behaved, we can partially differentiate both sides of the first of the two equations with respect to x, and partially differentiate both sides of the second equation with respect to y, to obtain

prefix partial differential of squared of u divided by prefix partial differential of of x squared equals prefix partial differential of squared of v divided by prefix partial differential of of x times prefix partial differential of of y and prefix partial differential of squared of v divided by prefix partial differential of of y times prefix partial differential of of x equals negative prefix partial differential of squared of u divided by prefix partial differential of of y squared full stop

(Here we have omitted the variables open x comma y close after each derivative, for simplicity.) For sufficiently well-behaved functions, the two partial derivatives

prefix partial differential of squared of v divided by prefix partial differential of of x times prefix partial differential of of y and prefix partial differential of squared of v divided by prefix partial differential of of y times prefix partial differential of of x

are equal; the order in which you partially differentiate with respect to x and y does not matter. Hence

equation sequence part 1 prefix partial differential of squared of u divided by prefix partial differential of of x squared equals part 2 prefix partial differential of squared of v divided by prefix partial differential of of x times prefix partial differential of of y equals part 3 prefix partial differential of squared of v divided by prefix partial differential of of y times prefix partial differential of of x equals part 4 negative prefix partial differential of squared of u divided by prefix partial differential of of y squared comma

which implies that

prefix partial differential of squared of u divided by prefix partial differential of of x squared plus prefix partial differential of squared of u divided by prefix partial differential of of y squared equals zero full stop

This equation for u is called Laplace’s equation. (The imaginary part v of f satisfies Laplace’s equation too.) It is named after the distinguished French mathematician and scientist Pierre-Simon Laplace (1749–1827), who studied the equation in his work on gravitational potentials.

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Pierre-Simon Laplace (1749–1827)
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This figure is a full-length portrait of Laplace as an older man, painted as a tribute after his death by the well-known contemporary artist Jean-Baptiste Paulin Guérin. Laplace is standing richly attired, clean-shaven, and wears a short white wig. At his side is a ceremonial sword, and round his shoulders a sumptuous velvet cloak. Some medals are visible on his chest, and in his right hand he is holding a velvet hat trimmed with fur. In the background are a pedestal with a classical bust, perhaps of a Greek goddess, and a table with a globe, a pair of compasses and some papers, presumably mathematical.

Pierre-Simon Laplace (1749–1827)

Laplace’s equation has proved to have huge importance to physics, with particular significance in fluid mechanics. It also has a key role in the subject of electrostatics. In that theory, it is known that the electrostatic potential cap v of x comma y at a point open x comma y close of a region without charge satisfies Laplace’s equation. It can be shown that cap v is the real part of some differentiable function f. Using these observations allows one to move between complex analysis and electrostatics: many of the theorems of complex analysis have important physical interpretations in electrostatics.