A Really Quick Derivation of The Cauchy-Riemann Equations

Here is a really quick derivation of the Cauchy-Riemann equations of complex analysis.

Consider a function of a complex variable, z, where z = x + iy, such that:

f(z) = u(z) + i v(z) = u(x+ iy) + i v(x+iy),

where u and v are real-valued functions.

An analytic function is one that is expressible as a power series in z.
That is,

f(z) = \sum_{n=0}^{\infty} a_{n} z^{n}, \quad a_{n} \in \mathbb{C}.


u(x+iy) + i v(x+iy) = \sum_{n=0}^{\infty} a_{n} (x+iy)^{n}.

We formally differentiate this equation as follows. First, differentiating with respect to x, we obtain

u_{x} + i v_{x} = \sum_{n=1}^{\infty} n a_{n} \left(x+iy\right)^{n-1}.

Differentiating with respect to y, we obtain

u_{y} + i v_{y} = i \sum_{n=1}^{\infty} n a_{n} \left(x + i y\right)^{n-1}.

Multiplying the latter equation by -i and equating to the first result, we obtain

-iu_{y} + v_{y} = \sum_{n=1}^{\infty} na_{n} \left(x+iy\right)^{n-1} = u_{x} + i v_{x}.

Comparing imaginary and real parts of these equations, we obtain

\boxed{u_{x} = v_{y}, \quad u_{y} = -v_{x}},

which are the famous Cauchy-Riemann equations.