Analytic Function and Singularity of a Function

Analytic Function and Singularity of a Function
Analytic Function
A mathematical function is said to be analytic at a point in a plane, if its value and its derivative has finite existence at that point.

Singularity of a Function
If at a point in plane, the value of function or its derivative is infinite, the function is said to be non-analytic at that point and such a point is called Singularity of the function.

Example : F (𝐬)=𝟐𝟓/(𝒔(𝒔+𝟏))

This function is analytic at all points except s = 0 and s= -1 which are its poles.

Generally we can say that , Poles of a function are its singularities (∵ F(S) = ꚙ)

Single valued Function
A function F(s) is said to be single valued function if it has one and only one value for each separate values of ‘s’.

Consider : F(s) = √25

Such a function is not said to be single valued.

In control system we assume that transfer function of system or the functions G(s) and H(s) are single valued