Transformation of the neighborhood of a branch point

[Eric_H]

Hi all,

I was trying the understand theory behind Fourier and Laplace Transform (especially in the context of control theory) by reading the book "Complex Variables and the Laplace Transform for Engineers" written by "Wilbur R. LePage".

In section 6-10 of the book the author touches on the application of multivalued functions in root locus and said "the portion of the real axis in the neighborhood of a branch point w_o on the real axis transforms into a system of radial 'spokes', as shown in Fig. 6-15."

I am confused about the statement as I thought the transformation of the neighborhood of a branch depends on the function we are considering. For instance in the earlier examples in the book, the sq root, cubic root (and in general n-th root I believe) do forms a star / radial 'spokes'. However, in the section 6-10, the function we are converning: w = A(s)H(s) is a ratio of polynomials which I assume may have different behaviour than a simple n-th root?

Any help is appreciated.

 

[FactChecker]

Any polynomial can be factored into a constant times integer powers of (z-a_i). More can be said about symmetry when the polynomial has all real coefficients. Although the factorization is different for different polynomials, the factors will always give branch points of the type shown.

PS. The behavior shown is only for a region near to the zero of the polynomial. In general, the "spokes" are curved with tangent lines at the zero point that look like the diagram.

 

[Eric_H]

Any polynomial can be factored into a constant times integer powers of (z-a_i). More can be said about symmetry when the polynomial has all real coefficients. Although the factorization is different for different polynomials, the factors will always give branch points of the type shown.

PS. The behavior shown is only for a region near to the zero of the polynomial. In general, the "spokes" are curved with tangent lines at the zero point that look like the diagram.

Thanks for your help. Do you mind explaining a bit more on how the factors of a polynomial will always give branch points of the type shown?
It does not look intuitive to me that
$$\Pi(z-a_i)$$
will generate such pattern.

In addition, the function w = A(s)H(s) is a ratio of polynomials, i.e. rational function, so I believe it should have this general form instead:
$$\frac{\Pi(z-a_i)}{\Pi(z-b_i)} $$
and therefore the effect of the denominator have to be taken into account too.

 

[FactChecker]

The zeros of the denominator are poles rather than branch points. They should be ignored for now, but they are very important for other considerations.
You should also include a constant multiplier as a factor. If all the coefficients are real, the multiplier will be real and any non-real zeros, a_i, will come in conjugate pairs. Same for the poles, b_i.
If all the a_is are distinct, then every zero will map the real line to a single line tangent to the real line at zero on the range plane. If there are n identical a_is, then you can consolidate them to a multiplier (z-a_k)ⁿ. That would give a branch point as described above. There would be n lines through a_k exactly evenly directed as described above that are mapped to curves through zero which are tangent to the real line.

 

[Eric_H]

The zeros of the denominator are poles rather than branch points. They should be ignored for now, but they are very important for other considerations.
You should also include a constant multiplier as a factor. If all the coefficients are real, the multiplier will be real and any non-real zeros, a_i, will come in conjugate pairs. Same for the poles, b_i.
If all the a_is are distinct, then every zero will map the real line to a single line tangent to the real line at zero on the range plane. If there are n identical a_is, then you can consolidate them to a multiplier (z-a_k)ⁿ. That would give a branch point as described above. There would be n lines through a_k exactly evenly directed as described above that are mapped to curves through zero which are tangent to the real line.

Thank you so much for the follow up, but I still cannot visualize why every zero will map the real line to a single line tangent to the real line at zero on the range plane, especially when the factors are multiplied together. The point about conjugate pair are roots is clear and is the result of complex conjugate root theorem. I am also curious about in which way the poles will affect the function / mapping.

Do you mind explaining a bit more about the above points or point me to some readings about them (e.g. some specific chapters of a book)?

Thanks a lot.