a Fourier series for . These trigonometric functions can themselves be expanded, using multiple angle formulae. Fluid flow Let the quantities and be the horizontal and vertical components of the velocity field of a steady incompressible, irrotational flow in two dimensions. The continuity condition for an incompressible flow is that and the condition that the flow be irrotational is that If we define the differential of a function by then the continuity condition is the integrability condition for this differential: the resulting function is called the stream function because it is constant along flow lines. The first derivatives of are given by and the irrotationality condition implies that satisfies the Laplace equation. The harmonic function that is conjugate to is called the velocity potential. The Cauchy–Riemann equations imply that Thus every analytic function corresponds to a steady incompressible, irrotational, inviscid fluid flow in the plane. The real part is the velocity potential, and the imaginary part is the stream function. Electrostatics According to Maxwell's equations, an electric field in two space dimensions that is independent of time satisfies and where is the charge density. The first Maxwell equation is the integrability condition for the differential so the electric potential may be constructed to satisfy The second of Maxwell's equations then implies that which is the Poisson equation. The Laplace equation can be used in three-dimensional problems in electrostatics and fluid flow just as in two dimensions. In three dimensions Fundamental solution A fundamental solution of Laplace's equation satisfies where the Dirac delta function denotes a unit source concentrated at the point . No function has this property: in fact it is a distribution rather than a function; but it can be thought of as a limit of functions whose integrals over space are unity, and whose support (the region where the function is non-zero) shrinks to a point (see weak solution). It is common to take a different sign convention for this equation than one typically does when defining fundamental solutions. This choice of sign is often convenient to work with because −Δ is a positive operator. The definition of the fundamental solution thus implies that, if the Laplacian of is integrated over any volume that encloses the source point, then The Laplace equation is unchanged under a rotation of coordinates, and hence we can expect that a fundamental solution may be obtained among solutions that only depend upon the