Zeros and poles

In complex analysis, a branch of mathematics, a pole is a special point where a complex-valued function behaves in an interesting way. It is the simplest kind of point where the function is not smooth or "regular." Technically, a point z₀ is a pole of a function f if it is a zero of the function 1/f and 1/f is smooth, or holomorphic, near that point.

A function f is called meromorphic in an area if, at every point in that area, either the function itself or its reciprocal, 1/f, is smooth nearby. When a function is meromorphic, zeros and poles are closely related: a zero of f is a pole of 1/f, and a pole of f is a zero of 1/f. This creates a special balance between zeros and poles, which is very important for studying meromorphic functions. For example, if a function is meromorphic everywhere in the complex plane, including a special point called the point at infinity, the total number of poles, counting how strong they are, equals the total number of zeros, also counting their strength.

Definitions

A function of a complex variable is holomorphic if it can be broken down into simple parts at every point in an area. This means it behaves nicely and follows special rules.

We call a zero of a function a point where the function’s value is zero. A pole is where the function’s opposite (one divided by the function) is zero. This helps us understand how functions behave in complex numbers.

At infinity

A function is called meromorphic at infinity if it behaves nicely outside a big circle and there is a number n such that the limit of the function divided by zⁿ as z goes to infinity exists and is not zero.

In this case, the point at infinity is a pole of order n if n is positive, and a zero of order |n| if n is negative.

For example, the function f(z) = 3/z has a pole of order 1 at z = 0, and a simple zero at infinity. Another function, f(z) = (z + 2)/((z - 5)²(z + 7)³), has poles of order 2 at z = 5 and order 3 at z = -7, with zeros at z = -2 and at infinity. The function f(z) = z has a single pole at infinity of order 1 and a zero at the origin.

All examples except the third are rational functions. For more information about zeros and poles of such functions, see Pole–zero plot § Continuous-time systems.

Function on a curve

The idea of zeros and poles can also be used for functions on special mathematical shapes called complex curves. Simple examples of these curves are the complex plane and the Riemann surface. We can study these functions by using special maps called charts.

When we look at a function on a compact curve — a special kind of closed shape — and the function behaves nicely everywhere on the curve, the number of zeros and poles it has is limited. Importantly, the total "strength" of the zeros matches the total "strength" of the poles. This idea is part of a bigger mathematical result called the Riemann–Roch theorem.