In mathematics, the Cauchy-Riemann 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 real-valued functions defined on an open subset of R². Then f is holomorphic if and only if u and v are differentiable and their partial derivatives satisfy the Cauchy-Riemann 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, z₀. We can essentially approach z₀ 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