Mathematics
The Cauchy–Riemann Equations and Holomorphic Functions
Quick fact
A complex function is holomorphic (complex-differentiable) exactly when its real and imaginary parts satisfy the Cauchy–Riemann equations: uₓ = vᵧ and uᵧ = -vₓ. These two simple partial differential equations are so powerful that they imply the function is infinitely differentiable and can be represented by a convergent power series.
Why this is interesting
You've seen derivatives for real functions, but what about functions that take in and output complex numbers? It turns out that the simple act of asking for a derivative in the complex world imposes a surprisingly strict condition that changes everything.