Define pole and zeros of a complex function
Answers
In mathematics, a zero of a function f(x) is a value asuch that f(a) = 0.
In complex analysis, zeros of holomorphic functionsand meromorphic functions play a particularly important role because of the duality between zeros and poles.
A function f of a complex variable z is meromorphic in the neighbourhood of a point {\displaystyle z_{0},} if either f or its reciprocal function 1/f is holomorphic in some neighbourhood of {\displaystyle z_{0}} (that is, if f or 1/f is differentiable in a neighbourhood of {\displaystyle z_{0}}). If {\displaystyle z_{0}} is a zero of 1/f , then it is a pole of f.
Thus a pole is a certain type of singularity of a function, nearby which the function behaves relatively regularly, in contrast to essential singularities, such as 0 for the logarithm function, and branch points, such as 0 for the complex square root function.