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

Jonathan Kent

Jonathan Kent

Answered question

2022-05-26

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?

Answer & Explanation

Hugo Bruce

Hugo Bruce

Beginner2022-05-27Added 10 answers

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 > 0 for zeros, and m n < 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.

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