what is the significance of poles and zero and wht is vector space?

Question

what is the significance of poles and zero and wht is vector space?

Answer 1

In complex analysis, a pole of a holomorphic function is a certain type of simple singularity that behaves like the singularity 1/z^n at z = 0. A pole of the function f(z) is a point z = a such that f(z) approaches infinity as z approaches a.



Zero or nothing is an important mathematical concept denoting absence which has facilitated the development of Maths and Science as we know them.



A vector space is a set of vectors of the same dimensions bound by the laws and properties common to all of them

Answer 2

The significance of the poles and zeros of a holomorphic function arise from the residue theorem. The basic idea is that the integral can be evaluated by adding the residues of the function at the poles.



Check out: http://mathworld.wolfram.com/ResidueTheorem.html

or any text on complex analysis.

Answer 3

1) z=a is called a pole of f(z) if |f(z)| approaches infinity as z approaches a.



2) z = a is called a zero of f(z) if f(a) = 0.



3) X is a vector space over a field F, if

(i) there is a binary operation + defined on X

such that (X, +) is a commutative group, and

(ii) there is a scalar multiplication operation .

which is defined for each element a of F and

x of X such that ax is in X, and satisfying

the distributive laws a(x + y) = ax + ay,

a(bx) = (ab)x.