Pole

From Conservapedia

In complex analysis a pole of a function is a singularity of the function that is well-behaved in a certain sense. More precisely we can say that a point z₀ is a pole of the function f if f has a singularity at z₀ but is holomorphic at z₀ for some positive integer n. The least such n is known as the order of the pole.

For example if we have a function f such that:, where z₁ is a constant, complex number and z is a complex variable, then it is impossible to calculate the value of f(z₁). This is so because we would have: . In this example the point z₁ is a singularity of f and the function has a pole of order 1 at z₁.

Anything divided by zero is often considered equivalent to infinity; however it is more illustrative to think of this as impossible to calculate, in other words, undefined (or indeterminate), a singularity.

Not all singularities correspond to poles. For example the function clearly has a singularity at 0. It can be seen from a power series expansion of the function that it has no pole at 0. This type of singularity is known as an essential singularity.