How can a function be analytic in the s-plane

[mech-eng]

Hi, I try to learn the subject of automatic control and there is an explanation which I cannot understand, probably because of background of mathematics. Here is the explanation. I would like to ask that how a function be analitc in the s-plane and would you like to explain it.

Source: Automatic Control Systems by Kuo

Thank you

 

[SteamKing]

View attachment 98948 Hi, I try to learn the subject of automatic control and there is an explanation which I cannot understand, probably because of background of mathematics. Here is the explanation. I would like to ask that how a function be analitc in the s-plane and would you like to explain it.

Source: Automatic Control Systems by Kuo

Thank you

Here is an article on the properties of analytic functions:

https://en.wikipedia.org/wiki/Analytic_function

 

[Jeff Rosenbury]

I think analytic would be opposed to not solvable analytically, i.e. numeric or stochastic methods were used to arrive at a solution.

 

[StephenBarton]

@jeff Resenbury
Can you please explain me what numeric or stochastic methods are used actually to solve it.

 

[FactChecker]

I think analytic would be opposed to not solvable analytically, i.e. numeric or stochastic methods were used to arrive at a solution.

In this context "analytic" means that it has a complex derivative in the complex plane, s. All polynomials are analytic in the entire complex plane and the quotient of polynomials is analytic everywhere in the complex plane except at the zeros of the denominator. So the function G(s) in the OP is analytic everywhere for s in the complex plane except at s = 0, -1, and -3.

 

[mech-eng]

In this context "analytic" means that it has a complex derivative in the complex plane, s. All polynomials are analytic in the entire complex plane and the quotient of polynomials is analytic everywhere in the complex plane except at the zeros of the denominator. So the function G(s) in the OP is analytic everywhere for s in the complex plane except at s = 0, -1, and -3.

What does complex derivative refer to?

Thank you.

 

[FactChecker]

What does complex derivative refer to?

Thank you.

The derivative of a complex-valued function, f, of a complex variable, z, at point z₀ is the complex number d, such that d = lim_z->z0( f(z)-f(z₀ ) / (z-z₀). The limit value, d, must be the same no matter what direction z approaches z₀ from.

When a function, f, has this property at each point in an area, the consequences are profound: If it has one derivative, then it also has all higher order derivatives. It has a convergent power series expansion. It's integral also has very special properties. I am sure that you will see a lot of this as you continue your studies.

 

[mech-eng]

The derivative of a complex-valued function, f, of a complex variable, z, at point z₀ is the complex number d, such that d = lim_z->z0( f(z)-f(z₀ ) / (z-z₀). The limit value, d, must be the same no matter what direction z approaches z₀ from.

When a function, f, has this property, the consequences are profound. If it has one derivative, then is has all higher order derivatives. It has a convergent power series expansion. It's integral also has very special properties. I am sure that you will see a lot of this as you continue your studies.

Is this topic belong to Complex Analysis Theory? In which books and at which titles of those books I can find this topic? Even I do not know enough Real Analysis [Calculus], first should I learn Real Analysis?

Thank you.

 

[FactChecker]

Is this topic belong to Complex Analysis Theory? In which books and at which titles of those books I can find this topic? Even I do not know enough Real Analysis [Calculus], first should I learn Real Analysis?

Thank you.

Yes. It is complex analysis. Real analysis would emphasize things that will not help much. I would look for a book in control laws that will summarize the basic complex analysis that is needed to understand control laws. If you need more, you can just read sections of complex analysis books. And the complex analysis books should be for engineers, not mathematicians.

PS. If you mean calculus when you say "Real Analysis", you do need some understanding of introductory undergraduate calculus. A Real Analysis book is very different.

 

[Jeff Rosenbury]

Can you please explain me what numeric or stochastic methods are used actually to solve it.

Not all differential equations have analytic solutions. When these problems need to be solved they are sometimes solved either numerically or stochastically.

To solve numerically, use a computer to solve small bits as if they were differential elements. Make sure to run an error analysis. An example would be weather modelling.

Stochastic modelling is sometimes done when the certainty of the model is in question. Insurance companies use them to set rates for example. Something like an accident rate might have a function found by curve fitting, but one can never tell if that function truly represents the data.

The point is that not all functions are "nice" for some value of nice. The definitions need to be abandoned or at least modified when a function is weird enough to not play by normal rules.