The Complex Derivative

Complex Analysis · Axiom Academy

The definition looks identical to real calculus — but because h can shrink to zero from any direction, it demands far more. Let f be defined on an open set containing z_0 . The derivative is the limit of the difference quotient as the complex increment h shrinks to 0 : h is complex — it can approach 0 along any path In the picture, h lives in its own copy of the plane centered at 0 . Watch it spiral inward: the angle of approach keeps changing while |h| collapses to zero. The derivative f'(z_0) exists only if the quotient settles on the same value no matter which direction h comes in on. Form the quotient and simplify — the algebra is the same as in real calculus: Every trace of h now sits in a single, harmless term: 2z + h . As the quotient lands on 2z — and crucially, h 's direction never appears, so it cannot change the answer. In the picture, h sweeps a small circle of approach directions on the left while the quotient value 2z + h traces the matching circle around the fixed point 2z on the right. As the circle shrinks, the value collapses onto 2z from all directions at once . The familiar power rule survives intact: f(z)=z^2 gives f'(z)=2z , just like on the real line. The same algebra makes every polynomial and e^ z differentiable too. Conjugation looks just as smooth — it simply flips the sign of the imaginary part. Yet form the quotient and the h refuses to cancel:

This is the written version of the interactive lesson above. See the full Complex Analysis course.