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Chapter 9: Problem 4

Using the argument principle, prove the Fundamental Theorem of Algebra.

Short Answer

Expert verified
The proof of the Fundamental Theorem of Algebra using the argument principle follows steps involving the organization of polynomial, selection of a suitable contour for integration, application of the argument principle and an exploration of behavior as the radius tends to infinity. In essence, it demonstrates that every non-constant polynomial must have at least a single complex root.

Step by step solution

01

Organizing the Polynomial

Consider a polynomial \(P(z) = a_nz^n + a_{n-1}z^{n-1} + ... + a_1z + a_0\) with \(a_n \neq 0\) and \(n \geq 1\). Divide through by \(a_n\), we get \(P(z) = z^n - beta_{n-1}z^{n-1} - ... - beta_1z - beta_0\)
02

Choose a Proper Contour for Integration

Choose a circle \(|z|=R\) centered at the origin as the contour for integration. Let \(f(z)=\frac{z^n}{P(z)}\), if \(z\) is not a zero of \(P(z)\).
03

Apply the Argument Principle

According to the argument principle, we have \(\frac{1}{2\pi i}\) times the integral of \(f'(z)/f(z)\) over the circle equals the number of zeros of \(P\) inside the circle, minus the number of poles of \(f\) inside the circle.
04

Indicate Behavior as Radius Tends to Infinity

As \(R\) tends to infinity, the integral of \(f'(z)/f(z)\) over the circle equals \(n\) times the integral of \(1/z\) over the circle, which is \(n\). However, the number of poles of \(f\) inside the circle does not change. Thus, the number of zeros of \(P\) must be \(n\), which means every non-constant polynomial has at least one complex root.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Argument Principle
The Argument Principle is a profound concept that connects complex analysis with counting zeros and poles of a function within a given region. In essence, the argument principle states that if you have a complex function, the change in its argument (or angle) as you trace a closed contour around it is connected to the number of zeros and poles inside that contour. This principle is particularly useful in the context of polynomials because it allows us to count how many zeros lie within a certain area.
The argument of a function is a measure of its angle in the complex plane. When dealing with a large circle contour, the contribution from terms with lower powers becomes negligible. Therefore, in the application of the argument principle, we often see its perfect alignment with robust mathematical theorems like the Fundamental Theorem of Algebra.
Complex Analysis
Complex Analysis is the rich field of mathematics dealing with complex numbers and functions of complex variables. This area of study allows mathematicians to work with two-dimensional numbers comprising real and imaginary parts.
When investigating polynomials and their properties, complex analysis provides very useful theorems. These include controlling the behavior of functions as they approach infinity or finding relationships between the derivative and the original function. For instance, choosing a circular contour in the complex plane and examining the behavior of the polynomial function, while effectively ignoring terms with lower powers, gives insight into the roots of the polynomial as outlined by complex analysis.
Polynomial Zeros
The zeros of a polynomial, sometimes called roots, are values of the variable for which the polynomial evaluates to zero. The Fundamental Theorem of Algebra guarantees that every non-linear polynomial equation has at least one complex root.
  • A polynomial of degree \( n \) has exactly \( n \) roots in the complex number set, counted with multiplicity.
  • By organizing the polynomial and using a contour large enough to contain all zeros, the calculations simplify, focusing mainly on the leading term.
  • The rest of the polynomial terms have a diminishing influence as the radius tends toward infinity.
In proving the Fundamental Theorem of Algebra, the application of these concepts inherently relies on counting these zeros using complex integration results.

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Most popular questions from this chapter

Let \(f(z)\) be analytic inside and on a simple closed contour \(C\) except for a finite number of poles inside \(C .\) Denote the zeros by \(z_{1}, \ldots, z_{n}\) (none of which lies on \(C\) ) and the poles by \(w_{1}, \ldots, w_{m} .\) If \(g(z)\) is analytic inside and on \(C\), prove that $$ \frac{1}{2 \pi i} \int_{C} g(z) \frac{f^{\prime}(z)}{f(z)} d z=\sum_{j=1}^{n} g\left(z_{j}\right)-\sum_{j=1}^{m} g\left(w_{j}\right) $$ where each zero and pole occurs as often in the sum as is required by its multiplicity.

Find the Laurent series for \(f(z)=\frac{z-12}{z^{2}+z-6}\) valid for (i) \(1<|z-1|<4\) (ii) \(|z-1|>1\) (iii) \(|z-1|<4\)

Let \(f(z)\) be analytic in the disk \(|z|1)\) except for a simple pole at a point \(z_{0},\left|z_{0}\right|=1\). Consider the expansion \(f(z)=a_{0}+a_{1} z+\) \(a_{2} z^{2}+\cdots\), and show that \(\lim _{n \rightarrow \infty}\left(a_{n} / a_{n+1}\right)=z_{0}\)

Find the principal part for the following Laurent series. (b) \(\frac{z^{2}}{z^{4}-1} \quad(0<|z+i|<\sqrt{2})\) (c) \(\frac{e^{z}}{z^{4}} \quad(|z|>0)\) (d) \(\frac{\sin z}{z^{4}} \quad(|z|>0)\) (e) \(\frac{1}{\tan ^{2} z}-\frac{1}{z^{2}} \quad(0<|z|<\pi / 2)\).

\(z \mid=R\) Let \(f\) be analytic on an open set \(D\), and \(f^{\prime}(a) \neq 0\) for some \(a \in D\). Show that $$ \int_{C} \frac{d z}{f(z)-f(a)}=\frac{2 \pi i}{f^{\prime}(a)} $$ where \(C\) is a sufficiently small circle centered at \(a\).
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