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

Consider a function of a complex variable, , where , such that:

,

where and are real-valued functions.

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

That is,

.

Then,

.

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

.

Differentiating with respect to , we obtain

.

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

.

Comparing imaginary and real parts of these equations, we obtain

,

which are the famous * Cauchy-Riemann equations*.