a holomorphic function
/ hol-oh-MOR-fik /
A holomorphic function is one that is complex-differentiable not just at a point but at EVERY point of an open set. This little upgrade — from 'differentiable here' to 'differentiable on an open region' — is the single most important definition in complex analysis, because everything good in the subject follows from it.
Concretely, f is holomorphic on an open set U if for every z_0 in U the difference quotient (f(z) - f(z_0))/(z - z_0) has a limit f'(z_0). Equivalently, writing f = u + i v with continuous partials, the Cauchy-Riemann equations u_x = v_y, u_y = -v_x hold throughout U. The phrase 'holomorphic at a point' is shorthand for 'holomorphic on some open disk around that point' — never on just the point alone, since openness is essential.
The reason this matters is the chain of free consequences. A holomorphic function is automatically infinitely differentiable; it equals a convergent power series in a neighborhood of each point; it preserves angles where its derivative is nonzero; its real and imaginary parts are harmonic; and globally it is astonishingly rigid (knowing it on a tiny arc determines it everywhere connected). No real-differentiable function carries that much hidden structure — this rigidity is the recurring theme of the whole subject.
Polynomials in z (like 3z^2 - z + 5) and e^z are holomorphic on the whole plane; 1/z is holomorphic on the open set C minus the origin; |z|^2 is holomorphic on no open set at all (CR holds only at the single point 0).
Holomorphy is a property on an OPEN set; differentiability at one isolated point does not count.
Being differentiable on an open set is the whole point — 'holomorphic at z_0' always implicitly means 'on a neighborhood of z_0'. A function differentiable at one point only is not holomorphic anywhere.