What is the general form of a rational function which has absolute value 1 on the circle | z | = 1? In particular, how are the zeros and poles related to each other?So, write R ( z ) = P ( z ) Q ( z ) , where P , Q are polynomials in z. The condition specifies that | R ( z ) | = 1 for all z such that | z | = 1. In other words, | P ( z ) | = | Q ( z ) | for all z such that | z | = 1. What can we say about P and Q?

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What is the general form of a rational function which has absolute value 1 on the circle       |    z      |    =  1? In particular, how are the zeros and poles related to each other?So, write   R  (  z  )  =                    P        (        z        )                    Q        (        z        )            , where   P  ,  Q are polynomials in   z. The condition specifies that       |    R  (  z  )      |    =  1 for all   z such that       |    z      |    =  1. In other words,       |    P  (  z  )      |    =      |    Q  (  z  )      |   for all   z such that       |    z      |    =  1. What can we say about   P and   Q?

Hugo Bruce · Accepted Answer

If a rational function   R satisfies   |  R  (  z  )  |  =  1 for   |  z  |  =  1, then the rational function  M  (  z  )  =  R  (  z  )  ⋅            R      (      1              /                    z        ¯            )        ¯  satisfies   M  (  z  )  =  1 for   |  z  |  =  1, therefore it is constant (a nonconstant rational function attains every value only finitely often), and   R satisfies  R  (  1      /        z    ¯    )  =  1      /              R      (      z      )        ¯    .Hence the poles and zeros of   R are related by reflection in the unit circle; if   ζ is a zero of order   k, then   1      /        ζ    ¯   is a pole of order   k and vice versa.Thus, if   (      a    n        )          0      ⩽      n      ⩽      N       are the distinct zeros and poles of   R in the unit disk, with orders       m    n   (      m    n    &amp;gt;  0 for zeros, and       m    n    &amp;lt;  0 for poles), and       a    0    =  0 [      m    0    =  0 is allowed], the product  B  (  z  )  =      z                  m        0              ⋅      ∏          n      =      1        N              (                        z          −                      a            n                                    1          −                                    a              n                        ¯                    z                    )                      m        n            is a rational function having exactly the same zeros and poles as   R, and also   |  B  (  z  )  |  =  1 for   |  z  |  =  1. So the quotient   R  (  z  )      /    B  (  z  ) is a rational function without zeros or poles, hence constant, and therefore  R  (  z  )  =  λ  ⋅  B  (  z  )for some   λ with   |  λ  |  =  1.

What is the general form of a rational function which has absolute value 1 on the circle <mrow c

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