In mathematics, the CauchyRiemann differential equations in complex analysis are two partial differential equations which provide a necessary and sufficient condition for a function to be holomorphic. Let f = u + iv be a function from an open subset of the complex numbers C to C, and regard u and v as realvalued functions defined on an open subset of R^{2}. Then f is holomorphic if and only if u and v are differentiable and their partial derivatives satisfy the CauchyRiemann equations, which are: and
It follows from the equations that u and v must be harmonic functions. The equations can therefore be seen as the conditions on a given pair of harmonic functions to come as real and imaginary parts of a complex_analytic function.
Derivation Consider a function f(z)=u(x,y)+i v(x,y) over C, and we wish to calculate its derivative at some point, z_{0}. We can essentially approach z_{0} along the real axis towards 0, or down the imaginary axis towards 0. If we take the first path: 
This is now in the form of two difference quotients, so now Taking the second path: 
Again, this is now in the form of two difference quotients, so Equating these two we get 
Equating real and imaginary parts, then 
∎
Polar representation Considering the polar representation z = re^{iθ}, the equations take the form
