Editing Bäcklund transform (section)

==The Cauchy–Riemann equations==
{{Further|Cauchy–Riemann equations}}
The prototypical example of a Bäcklund transform is the [[Cauchy–Riemann equations|Cauchy–Riemann system]]

:<math>u_x=v_y, \quad u_y=-v_x,\,</math>

which relates the real and imaginary parts <math>u</math> and <math>v</math> of a [[holomorphic function]]. This first order system of partial differential equations has the following properties.

# If <math>u</math> and <math>v</math> are solutions of the Cauchy–Riemann equations, then <math>u</math> is a solution of the [[Laplace equation]] <br /><math>u_{xx} + u_{yy} = 0</math> <br />(i.e., a [[harmonic function]]), and so is <math>v</math>. This follows straightforwardly by differentiating the equations with respect to <math>x</math> and <math>y</math> and using the fact that <br /><math>u_{xy}=u_{yx}, \quad v_{xy}=v_{yx}.\,</math>
# Conversely if <math>u</math> is a solution of Laplace's equation, then there exist functions <math>v</math> which solve the Cauchy–Riemann equations together with <math>u</math>.
Thus, in this case, a Bäcklund transformation of a harmonic function is just a [[conjugate harmonic function]]. The above properties mean, more precisely, that Laplace's equation for <math>u</math> and Laplace's equation for <math>v</math> are the [[integrability condition]]s for solving the Cauchy–Riemann equations.

These are the characteristic features of a Bäcklund transform. If we have a partial differential equation in <math>u</math>, and a Bäcklund transform from <math>u</math> to <math>v</math>, we can deduce a partial differential equation satisfied by <math>v</math>.

This example is rather trivial, because all three equations (the equation for <math>u</math>, the equation for <math>v</math> and the Bäcklund transform relating them) are linear. Bäcklund transforms are most interesting when just one of the three equations is linear.