Problem 3 - Complex Analysis for Mathematics and Engineering

Question

Let \(g(z)=z^{5}+4 z-15\) (a) Show that there are no zeros in \(D_{1}(0)\). (b) Show that there are five zeros in \(D_{2}(0)\). Hint: Consider \(f(z)=-z^{5}\). Remark: A factorization of the polynomial using numerical approximations for the coefficients is \((z-1.546)\left(z^{2}-1.340 z+2.857\right)\left(z^{2}+2.885 z+3.397\right)\)

Step-by-Step Solution

Answer

(a) No zeros in \(D_1(0)\); (b) Five zeros in \(D_2(0)\).

Step 1: Analyzing the Function Inside D1(0)

To determine the presence of zeros, we will analyze the function \(g(z)\) inside the disk \(D_1(0)\), which is the set of all points such that \(|z| < 1\). Calculate the maximum value of \(|g(z)|\) on the boundary \(|z| = 1\).

Step 2: Estimating Maximum on the Boundary

If \(|z| = 1\), then \[ \begin{align*} \left|g(z)\right| &= \left|z^5 + 4z - 15\right| \ &\geq \left|15\right| - \left|z^5 + 4z\right| \ &= 15 - (|1| + 4|1|) = 15 - 5 = 10. \end{align*} \] Since this value is positive, \(g(z) eq 0\) on \(D_1(0)\). Thus, \(g(z)\) has no zeros in \(D_1(0)\).

Step 3: Applying Rouche's Theorem on D2(0)

Next, to show that there are five zeros inside \(D_2(0)\), use Rouche's Theorem. Consider \(f(z) = -z^5\) and \(h(z) = 4z - 15\). On the boundary \(|z| = 2\), \[|f(z)| = |z|^5 = 32\, \text{and}\, |h(z)| = |4z - 15| \leq 8 + 15 = 23.\]Since \(|f(z)| > |h(z)|\) on the contour, Rouche's Theorem tells us \(g(z)\) and \(f(z)\) have the same number of zeros in \(D_2(0)\).

Step 4: Counting Zeros

From Step 3, since \(-z^5\) has 5 zeros at the origin in \(D_2(0)\), \(g(z)\) also has 5 zeros within this disk. The factorization confirms zero positions but was not necessary by the hint provided.

Key Concepts

Rouche's Theorem

Rouche's Theorem is a powerful tool in complex analysis, especially useful for counting zeros of analytic functions within specific contours in the complex plane. This theorem states that if two analytic functions, let’s call them \( f(z) \) and \( h(z) \), are defined on a disk and if the inequality \( |f(z) + h(z)| > |h(z)| \) holds on the boundary of that disk, then \( f(z) \) and \( f(z) + h(z) \) have the same number of zeros inside that disk.

To apply Rouche's Theorem effectively:

• Choose an analytic function \( f(z) \) that has known zero counts within the disk.
• Choose an appropriate \( h(z) \) such that \( g(z) = f(z) + h(z) \) represents the function you want to analyze.
• Verify the inequality on the boundary of the disk. It usually means showing \( |f(z)| > |h(z)| \) on that boundary.

In the exercise, when analyzing the disk \( D_2(0) \), we see \( f(z) = -z^5 \) already has an understanding of zero locations. Since it dominates \( h(z) = 4z - 15 \) on the boundary, Rouche's Theorem confirms the zero scenario for the polynomial \( g(z) \).

Rouche's Theorem provides a bridge to handling complicated polynomial forms by breaking them down into manageable parts.

Zeros of Polynomials

Finding the zeros of a polynomial is a fundamental problem in both algebra and complex analysis. A polynomial zero is where the polynomial evaluates to zero when substituted with a particular value of \( z \).

When counting zeros of polynomials:

• Consider factors of the polynomial form. Any time the polynomial can be factored into terms like \( (z - a) \), it signifies a zero at \( z = a \).
• Zeros might be real, complex, or a combination thereof based on polynomial degree and coefficients.
• For complex polynomials, Euler’s formula and trigonometric identities often assist in simplifying complex number evaluations.

The degree of a polynomial provides an important hint regarding the total number of zeros, considering multiplicity in the count. In the exercise, \( g(z) = z^5 + 4z - 15 \) is a polynomial of degree five, which anticipates a total of five zeros (some may be identical if they have multiplicity greater than one).

Zeros help determine the behavior of the polynomial on the complex plane, making their identification crucial for interpreting any physical or theoretical model described by such a polynomial.

Analytic Functions

Analytic functions, or holomorphic functions, are functions that are complex differentiable at all points in their domain. Such functions boast numerous fascinating properties, primarily owing to their differentiability, which extends beyond mere differentiation in real analysis.

Key characteristics of analytic functions include:

• They can be locally expressed as a power series, equipping them with unmatched smoothness and continuity.
• Everywhere they are defined, they maintain an open domain without breaks or sharp corners, which allows consistent and predictable behavior.
• Analytic functions obey the Cauchy-Riemann equations, ensuring their differentiability at least in an infinitesimal neighborhood.

In complex analysis, these functions are essential when dealing with integral formulas and theorems, such as the Cauchy Integral Theorem and the aforementioned Rouche's Theorem.

In discussing \( g(z) = z^5 + 4z - 15 \), which is a polynomial, this function is analytic everywhere in the complex plane. This analyticity allows applying sophisticated theorems like Rouche's to reveal zero placements which are integral to solving higher-level polynomial equations elegantly.