the order of a zero
When an analytic function vanishes at a point, it can do so gently or forcefully — touching zero and bouncing away, or pressing down hard and flattening against it. The order of the zero measures exactly how hard. It is the smallest power of (z - z_0) you can factor out, equivalently the number of derivatives of f that vanish at z_0 before one finally does not. A simple zero (order 1) is a clean crossing; a higher-order zero is a tangency where the function and several of its derivatives all vanish together.
Precisely: z_0 is a zero of order m if f(z_0) = 0, f'(z_0) = 0, ..., f^(m-1)(z_0) = 0, but f^(m)(z_0) is not 0. Equivalently, the Taylor series at z_0 begins at the m-th term: f(z) = a_m (z - z_0)^m + a_(m+1)(z - z_0)^(m+1) + ..., with a_m not equal to 0. Equivalently again, f(z) = (z - z_0)^m g(z) with g analytic and g(z_0) not equal to 0. To find the order in practice, expand the Taylor series and read off the index of the first surviving coefficient — that index is m. For example, 1 - cos z = z^2/2 - z^4/24 + ... starts at z^2, so it has a zero of order 2 at the origin.
Order is the bookkeeping device that makes counting zeros honest. In the fundamental theorem of algebra a degree-n polynomial has n zeros only when each is counted according to its order. In the argument principle, a contour integral counts zeros with their orders. And the order controls local behaviour under mappings: near a zero of order m, the function behaves like (z - z_0)^m, so it wraps angles by a factor of m and is locally m-to-1. The order is therefore not a curiosity but the quantitative soul of how and how strongly a function vanishes.
f(z) = z^3 (z - 1) has a zero of order 3 at z = 0 (since z^3 factors out) and a simple zero of order 1 at z = 1. For sin^2(z) at z = 0: sin z = z - ... has order 1, so sin^2(z) starts at z^2 and has a zero of order 2. Orders multiply for products and add when you square.
The order tells you how many factors of (z - z_0) hide in f; for a product, the orders at a shared zero add.
Order is always a finite positive integer for a nonzero analytic function — it cannot be a fraction or infinite. An 'infinite-order zero' would mean every Taylor coefficient vanishes, forcing the function to be identically zero on its disk.