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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.