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Introduction to complex analysis

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# 2.4 Laplace’s equation and electrostatics

The Cauchy–Riemann equations for a differentiable function tell us that

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

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

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

which implies that

This equation for is called Laplace’s equation. (The imaginary part of 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.

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 at a point of a region without charge satisfies Laplace’s equation. It can be shown that is the real part of some differentiable function . 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.