Rational Functions: Analysis & Agreement

• Pere Callahan
In summary, it is possible for two rational functions to coincide on the unit circle but not everywhere. However, if the poles of both functions are removed and they are analytic on a connected domain where they also agree on the unit circle, then they must coincide on the entire domain according to the uniqueness principle. The same does not apply if only the modulus of the functions is the same on the unit circle.
Pere Callahan
Hi,

I was wondering whether two rational functions f,g whch coincide on the unit circle actually coincide on all of C.

I would say yes. Let D be the set of all complex numbers with the poles of both f and g removed (let's assume there are no poles on the unit circle). This is then open and connected, hence a domain and f and g are analytic there. Moreover they agree on the unit circle which is a set with at least one nonisolated point (in fact all points are nonisolated) and which lies in D, so the uniqueness principle implies that f and g agree on D.

But the poles have to be the same as well. For if w is a pole of f but not of g then the limit of f as z approaches w is infinity and must be the same as the limit of g as w approaches infinty, because a neighbourhood of z is contained in D.

Is this correct?

thanks

Anyone?

Ok, I think my argument from above is correct.
But what if we only know that the two rational functions' modulus is the same on the unit circle?

Do they still have to coincide everywhere? I don't know how to adopt my previous reasining because the modulus is not an analytic function..

Thanks.

To your original question, the answer is yes. You can use the values of f-g on any convergent sequence to a point on the circle to expand in a series about that point - in that sequence the coefficients are all zero. QED

To your second question its not true - consider z and -z for example.

1. What is a rational function?

A rational function is a mathematical function that can be expressed as the ratio of two polynomial functions. It is typically written in the form f(x) = P(x)/Q(x), where P(x) and Q(x) are polynomial functions and Q(x) is not equal to zero.

2. How do you analyze a rational function?

To analyze a rational function, you can start by finding its domain and identifying any vertical or horizontal asymptotes. Then, you can determine the x and y intercepts and plot them on a graph. Next, you can analyze the behavior of the function near the asymptotes and determine its end behavior. Finally, you can use the graph to find the range and any important features of the function.

3. What is meant by agreement in rational functions?

In rational functions, agreement refers to the condition where the numerator and denominator of the function have the same degree. This means that the highest power of x in the numerator is equal to the highest power of x in the denominator. When this condition is met, the function can be simplified and its behavior can be easily determined.

4. How do you find the vertical asymptote of a rational function?

The vertical asymptote of a rational function can be found by setting the denominator of the function equal to zero and solving for x. The resulting value(s) of x will be the vertical asymptote(s) of the function. If the function is in factored form, the vertical asymptotes will be the values that make one of the factors equal to zero.

5. Can a rational function have more than one horizontal asymptote?

Yes, a rational function can have more than one horizontal asymptote. This occurs when the degree of the numerator is less than the degree of the denominator by more than one. In this case, the function will have two or more horizontal asymptotes, depending on the difference in degrees between the numerator and denominator.

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