Editing Laplace's equation (section)

==In two dimensions==
Laplace's equation in two independent variables in rectangular coordinates has the form 
<math display="block">\frac{\partial^2\psi}{\partial x^2} + \frac{\partial^2\psi}{\partial y^2} \equiv \psi_{xx} + \psi_{yy} = 0.</math>
===Analytic functions===
<!-- This section is linked from [[Complex analysis]] -->
The real and imaginary parts of a complex [[analytic function]] both satisfy the Laplace equation. That is, if {{math|1=''z'' = ''x'' + ''iy''}}, and if 
<math display="block">f(z) = u(x,y) + iv(x,y),</math>
then the necessary condition that {{math|''f''(''z'')}} be analytic is that {{math|''u''}} and {{mvar|''v''}} be differentiable and that the [[Cauchy–Riemann equations]] be satisfied:
<math display="block">u_x = v_y, \quad v_x = -u_y.</math>
where {{math|''u<sub>x</sub>''}} is the first partial derivative of {{math|''u''}} with respect to {{mvar|x}}.
It follows that 
<math display="block">u_{yy} = (-v_x)_y = -(v_y)_x = -(u_x)_x.</math>
Therefore {{math|''u''}} satisfies the Laplace equation. A similar calculation shows that {{math|''v''}} also satisfies the Laplace equation. 
Conversely, given a harmonic function, it is the real part of an analytic function, {{math|''f''(''z'')}} (at least locally). If a trial form is
<math display="block">f(z) = \varphi(x,y) + i \psi(x,y),</math>
then the Cauchy–Riemann equations will be satisfied if we set
<math display="block">\psi_x = -\varphi_y, \quad \psi_y = \varphi_x.</math>
This relation does not determine {{math|''ψ''}}, but only its increments:
<math display="block">d \psi = -\varphi_y\, dx + \varphi_x\, dy.</math>
The Laplace equation for {{math|''φ''}} implies that the integrability condition for {{math|''ψ''}} is satisfied:
<math display="block">\psi_{xy} = \psi_{yx},</math>
and thus {{math|''ψ''}} 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 {{mvar|r}} and {{mvar|θ}} are polar coordinates and
<math display="block">\varphi = \log r,</math>
then a corresponding analytic function is
<math display="block">f(z) = \log z = \log r + i\theta.</math>

However, the angle {{mvar|θ}} 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{{citation needed|date=July 2020}}.

There is an intimate connection between power series and [[Fourier series]]. If we expand a function {{math|''f''}} in a power series inside a circle of radius {{mvar|R}}, this means that
<math display="block">f(z) = \sum_{n=0}^\infty c_n z^n,</math>
with suitably defined coefficients whose real and imaginary parts are given by 
<math display="block">c_n = a_n + i b_n.</math>
Therefore
<math display="block">f(z) = \sum_{n=0}^\infty \left[ a_n r^n \cos n \theta - b_n r^n \sin n \theta\right] + i \sum_{n=1}^\infty \left[ a_n r^n \sin n\theta + b_n r^n \cos n \theta\right],</math>
which is a Fourier series for {{math|''f''}}.  These trigonometric functions can themselves be expanded, using [[De Moivre's formula#Formulas for cosine and sine individually|multiple angle formulae]].

===Fluid flow===
{{Main|Laplace equation for irrotational flow}}
Let the quantities {{math|''u''}} and {{math|''v''}} 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 
<math display="block">u_x + v_y=0,</math>
and the condition that the flow be irrotational is that
<math display="block">\nabla \times \mathbf{V} = v_x - u_y = 0.</math>
If we define the differential of a function {{math|''ψ''}} by
<math display="block">d \psi = u \, dy - v \, dx,</math>
then the continuity condition is the integrability condition for this differential: the resulting function is called the [[stream function]] because it is constant along [[Streamlines, streaklines and pathlines|flow lines]]. The first derivatives of {{math|''ψ''}} are given by
<math display="block">\psi_x = -v, \quad \psi_y=u,</math>
and the irrotationality condition implies that {{math|''ψ''}} satisfies the Laplace equation. The harmonic function {{math|''φ''}} that is conjugate to {{math|''ψ''}} is called the [[velocity potential]]. The Cauchy–Riemann equations imply that
<math display="block">\varphi_x=\psi_y=u, \quad \varphi_y=-\psi_x=v.</math>
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 {{math|(''u'', ''v'')}} in two space dimensions that is independent of time satisfies
<math display="block">\nabla \times (u,v,0) = (v_x -u_y)\hat{\mathbf{k}} = \mathbf{0},</math>
and
<math display="block">\nabla \cdot (u,v) = \rho,</math>
where {{math|''ρ''}} is the charge density. The first Maxwell equation is the integrability condition for the differential
<math display="block">d \varphi = -u\, dx -v\, dy,</math>
so the electric potential {{math|''φ''}} may be constructed to satisfy
<math display="block">\varphi_x = -u, \quad \varphi_y = -v.</math>
The second of Maxwell's equations then implies that 
<math display="block">\varphi_{xx} + \varphi_{yy} = -\rho,</math>
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.