of Laplace's equation are called harmonic functions; they are all analytic within the domain where the equation is satisfied. If any two functions are solutions to Laplace's equation (or any linear homogeneous differential equation), their sum (or any linear combination) is also a solution. This property, called the principle of superposition, is very useful. For example, solutions to complex problems can be constructed by summing simple solutions. In two dimensions Laplace's equation in two independent variables in rectangular coordinates has the form Analytic functions The real and imaginary parts of a complex analytic function both satisfy the Laplace equation. That is, if , and if then the necessary condition that be analytic is that and be differentiable and that the Cauchy–Riemann equations be satisfied: where is the first partial derivative of with respect to . It follows that Therefore satisfies the Laplace equation. A similar calculation shows that also satisfies the Laplace equation. Conversely, given a harmonic function, it is the real part of an analytic function, (at least locally). If a trial form is then the Cauchy–Riemann equations will be satisfied if we set This relation does not determine , but only its increments: The Laplace equation for implies that the integrability condition for is satisfied: and thus may be defined by a line integral. The integrability condition and Stokes' theorem implies that the value of the line integral connecting two points is independent of the path. The resulting pair of solutions of the Laplace equation are called conjugate harmonic functions. This construction is only valid locally, or provided that the path does not loop around a singularity. For example, if and are polar coordinates and then a corresponding analytic function is However, the angle is single-valued only in a region that does not enclose the origin. The close connection between the Laplace equation and analytic functions implies that any solution of the Laplace equation has derivatives of all orders, and can be expanded in a power series, at least inside a circle that does not enclose a singularity. This is in sharp contrast to solutions of the wave equation, which generally have less regularity. There is an intimate connection between power series and Fourier series. If we expand a function in a power series inside a circle of radius , this means that with suitably defined coefficients whose real and imaginary parts are given by Therefore which is