An analytic function's behaviour in one tiny patch pins down all the rest
Analytic functions are the most rigid kind in mathematics. On a connected domain, if one of them hits zero at a cluster of points that pile up around some location, it must be zero absolutely everywhere. Knowing such a function near a single point, through its derivatives there, fixes it across its whole domain.
The defining idea is simple to state. A function is analytic at a point if, somewhere around that point, it equals a convergent power series, an endless polynomial built from its derivatives at that centre, known as a Taylor series. The coefficients of that series, and so the local behaviour, are all determined by what happens at one spot.
Real and complex numbers treat the idea differently. Any analytic function is smooth, meaning it can be differentiated as many times as you like. On the real line the reverse fails: some perfectly smooth functions are not analytic, including any smooth bump that is nonzero somewhere but vanishes outside a bounded region. In the complex plane, though, a function that can be differentiated once at every point of an open set is automatically analytic there, so complex analysts treat analytic and holomorphic as synonyms. Engineers in signal processing have their own term, calling such a complex function an analytic signal.
Familiar examples abound. Every polynomial qualifies, being its own series expansion, and the exponential function's series converges for every input. Sine and cosine are analytic across their domains, as is the natural logarithm wherever a single branch is chosen. Heavier machinery fits too: the gamma function away from zero and the negative integers, and the Riemann zeta function apart from one pole.
Functions stitched together from different formulas in different regions usually lose analyticity along the seams. Taking the complex conjugate is another failure case: it is not complex analytic on any open set, even though on the real line it reduces to the identity function, which is real analytic.
Source: Analytic function
