## 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,

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

Then,

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

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.