Complex Analysis - Rational Functions

Thread starter ilikegroupreps
Start date May 8, 2013
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Analysis Complex Complex analysis Functions Rational Rational functions

In summary, complex analysis is a branch of mathematics that studies functions of complex numbers and their properties such as continuity and differentiability. Rational functions, which are expressed as the ratio of two polynomials, play a significant role in complex analysis as they can approximate more complex functions and provide insight into the behavior of other types of functions in the complex plane. The poles and zeros of a rational function are the points where the denominator and numerator are equal to 0, respectively, and can affect the function's behavior. Graphically, rational functions are represented as curves or surfaces in the complex plane, with the zeros and poles serving as points of intersection and the curvature and asymptotic lines providing information about the function's behavior.

May 8, 2013

ilikegroupreps

Homework Statement

I'm studying for my final exam and came across this problem:
Let f and g be entire analytic functions and |f(z)|<|g(z)| when |z|>1. Show that f/g is a rational function.

The Attempt at a Solution

I really have no clue where to go :(

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May 10, 2013

christoff

Well, you know that since |f(z)|<|g(z)| on |z|>1, f(z)/g(z) is bounded on |z|>1. Does this tell you anything?

Related to Complex Analysis - Rational Functions

1. What is complex analysis?

Complex analysis is a branch of mathematics that deals with the study of functions of complex numbers. It involves the analysis of properties such as continuity, differentiability, and integrability of these functions in the complex plane.

2. What are rational functions?

Rational functions are functions that can be expressed as the ratio of two polynomials. In other words, they are functions of the form f(x) = p(x)/q(x), where p(x) and q(x) are polynomials.

3. What is the significance of rational functions in complex analysis?

Rational functions play a crucial role in complex analysis as they can be used to approximate more complex functions and can also help in understanding the behavior of other types of functions in the complex plane.

4. What are the poles and zeros of a rational function?

In a rational function f(x) = p(x)/q(x), the poles are the values of x for which q(x) is equal to 0, while the zeros are the values of x for which p(x) is equal to 0. These points are important as they can affect the behavior of the function in the complex plane.

5. How are rational functions graphically represented in the complex plane?

Rational functions are typically represented as curves or surfaces in the complex plane. The zeros and poles of the function can be seen as points where the curve or surface intersects the x-axis. The behavior of the function can also be determined by looking at the curvature and asymptotic lines of the graph.