Editing Cauchy–Riemann equations (section)

=== Other representations ===
Other representations of the Cauchy–Riemann equations occasionally arise in other [[coordinate system]]s. If ({{EquationNote|1a}}) and ({{EquationNote|1b}}) hold for a differentiable pair of functions ''u'' and ''v'', then so do
<math display="block">
  \frac{\partial u}{\partial n} =  \frac{\partial v}{\partial s},\quad
  \frac{\partial v}{\partial n} = -\frac{\partial u}{\partial s}
</math>

for any coordinate system {{math|(''n''(''x'', ''y''), ''s''(''x'', ''y''))}} such that the pair <math display="inline">(\nabla n,\nabla s)</math> is [[orthonormal]] and [[orientation (space)|positively oriented]]. As a consequence, in particular, in the system of coordinates given by the polar representation <math>z = r e^{i\theta}</math>, the equations then take the form
<math display="block">
  {\partial u \over \partial r} =  {1 \over r}{\partial v \over \partial\theta},\quad
  {\partial v \over \partial r} = -{1 \over r}{\partial u \over \partial\theta}.
</math>

Combining these into one equation for {{math|''f''}} gives
<math display="block">{\partial f \over \partial r} = {1 \over ir}{\partial f \over \partial\theta}.</math>

The inhomogeneous Cauchy–Riemann equations consist of the two equations for a pair of unknown functions {{math|''u''(''x'', ''y'')}} and {{math|''v''(''x'', ''y'')}} of two real variables
<math display="block">\begin{align}
  \frac{\partial u}{\partial x} - \frac{\partial v}{\partial y} &= \alpha(x, y) \\[4pt]
  \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} &= \beta(x, y)
\end{align}</math>

for some given functions {{math|α(''x'', ''y'')}} and {{math|β(''x'', ''y'')}} defined in an open subset of '''R'''<sup>2</sup>. These equations are usually combined into a single equation
<math display="block">\frac{\partial f}{\partial\bar{z}} = \varphi(z,\bar{z})</math>
where ''f'' = ''u'' + i''v'' and ''𝜑'' = (''α'' + i''β'')/2.

If ''𝜑'' is [[continuously differentiable|''C''<sup>''k''</sup>]], then the inhomogeneous equation is explicitly solvable in any bounded domain ''D'', provided ''𝜑'' is continuous on the [[closure (topology)|closure]] of ''D''. Indeed, by the [[Cauchy integral formula]],
<math display="block">f\left(\zeta, \bar{\zeta}\right) = \frac{1}{2\pi i} \iint_D \varphi\left(z, \bar{z}\right) \, \frac{dz\wedge d\bar{z}}{z - \zeta}</math>
for all ''ζ'' ∈ ''D''.