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In [[complex analysis]], the '''Cauchy-Riemann equations''' are two [[partial derivative|partial]] [[differential equation]]s that a [[complex number|complex]] [[function]] must satisfy in some region of the [[complex plane]] for it to be [[analytic]] in that region.<ref name=MathematicalMethods>K.F. Riley, M.P. Hobson, S.J. Bence, ''Mathematical Methods for Physics and Engineering'', Cambridge University Press, 3<sup>rd</sup> ed., 2006</ref><ref name=wolfram>[http://mathworld.wolfram.com/Cauchy-RiemannEquations.html Cauchy-Riemann Equations] from mathworld.wolfram.com</ref> This has important implications for the [[differentiation|differentiability]] of complex functions. If we express a complex number ''z'' in terms of two [[real number]]s, ''x'' and ''y'', through ''z''=''x''+''iy'', then a complex function, ''f(z)'', can be expressed in terms of two functions of ''x'' and ''y'', ''u(x,y)'' and ''v(x,y)'', as,
:<math>
f(z) = u(x,y) + iv(x,y)
</math>
The Cauchy-Riemann equations can then be expressed in terms of ''u(x,y)'' and ''v(x,y)'' as,<ref name=wolfram/>
:<math>
\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \qquad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}
</math>
They are named after [[Augustin-Louis Cauchy]] and [[Bernhard Riemann]].
==Derivation==
When defining the [[derivative]] of a real function, it is done through a [[limit (mathematics)|limit]]. The same method can be used for complex differentiation, defining it as,<ref name=MathematicalMethods/>
:<math>
L = \lim_{\Delta z \to 0} \frac{f(z+\Delta z) - f(z)}{\Delta z}
</math>
However we require the derivative and so this limit to be unique. But in the complex plane we can approach the limit in many different ways (e.g. along the real axis or the imaginary axis among others). These must all give the same result. If Δ''z''=Δ''x''+''i''Δ''y'' and expressing ''f'' in terms of ''u(x,y)'' and ''v(x,y)'' as before then the limit becomes,
:<math>
L = \lim_{\Delta x, \Delta y \to 0}
\left[ \frac{u(x+\Delta x, y+\Delta y) + v(x+\Delta x, y+\Delta y) - u(x,y) - v(x,y)}{\Delta x + i\Delta y} \right]
</math>
Consider the cases of Δ''z'' being purely real or purely imaginary, corresponding to taking the limit along wither the real or imaginary axes. If Δ''z'' is real then Δ''y'' is zero meaning,
:<math>
L = \lim_{\Delta x \to 0} \left[\frac{u(x+\Delta x,y) - u(x,y)}{\Delta x} + i \frac{v(x+\Delta x,y) - v(x,y)}{\Delta x} \right]
=\frac{\partial u}{\partial x} + i \frac{\partial v}{\partial x}
</math>
If instead the case of Δ''z'' being imaginary (Δ''x'' is zero) is considered then,
:<math>
L = \lim_{\Delta y \to 0} \left[\frac{u(x,y+\Delta y) - u(x,y)}{i\Delta y} + i \frac{v(x,y+\Delta y) - v(x,y)}{i\Delta y} \right]
=\frac{1}{i} \frac{\partial u}{\partial y} + \frac{\partial v}{\partial y}
</math>
For ''f'' to be differentiable at a point ''z'' these two must be identical. Comparing real and imaginary parts gives the Cauchy-Riemann equations,
:<math>
\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \qquad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}
</math>
==References==
{{reflist}}
==See also==
*[[Complex analysis]]
*[[Complex number]]
[[Category:Mathematics]]
[[Category:Complex analysis]]
:<math>
f(z) = u(x,y) + iv(x,y)
</math>
The Cauchy-Riemann equations can then be expressed in terms of ''u(x,y)'' and ''v(x,y)'' as,<ref name=wolfram/>
:<math>
\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \qquad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}
</math>
They are named after [[Augustin-Louis Cauchy]] and [[Bernhard Riemann]].
==Derivation==
When defining the [[derivative]] of a real function, it is done through a [[limit (mathematics)|limit]]. The same method can be used for complex differentiation, defining it as,<ref name=MathematicalMethods/>
:<math>
L = \lim_{\Delta z \to 0} \frac{f(z+\Delta z) - f(z)}{\Delta z}
</math>
However we require the derivative and so this limit to be unique. But in the complex plane we can approach the limit in many different ways (e.g. along the real axis or the imaginary axis among others). These must all give the same result. If Δ''z''=Δ''x''+''i''Δ''y'' and expressing ''f'' in terms of ''u(x,y)'' and ''v(x,y)'' as before then the limit becomes,
:<math>
L = \lim_{\Delta x, \Delta y \to 0}
\left[ \frac{u(x+\Delta x, y+\Delta y) + v(x+\Delta x, y+\Delta y) - u(x,y) - v(x,y)}{\Delta x + i\Delta y} \right]
</math>
Consider the cases of Δ''z'' being purely real or purely imaginary, corresponding to taking the limit along wither the real or imaginary axes. If Δ''z'' is real then Δ''y'' is zero meaning,
:<math>
L = \lim_{\Delta x \to 0} \left[\frac{u(x+\Delta x,y) - u(x,y)}{\Delta x} + i \frac{v(x+\Delta x,y) - v(x,y)}{\Delta x} \right]
=\frac{\partial u}{\partial x} + i \frac{\partial v}{\partial x}
</math>
If instead the case of Δ''z'' being imaginary (Δ''x'' is zero) is considered then,
:<math>
L = \lim_{\Delta y \to 0} \left[\frac{u(x,y+\Delta y) - u(x,y)}{i\Delta y} + i \frac{v(x,y+\Delta y) - v(x,y)}{i\Delta y} \right]
=\frac{1}{i} \frac{\partial u}{\partial y} + \frac{\partial v}{\partial y}
</math>
For ''f'' to be differentiable at a point ''z'' these two must be identical. Comparing real and imaginary parts gives the Cauchy-Riemann equations,
:<math>
\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \qquad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}
</math>
==References==
{{reflist}}
==See also==
*[[Complex analysis]]
*[[Complex number]]
[[Category:Mathematics]]
[[Category:Complex analysis]]