Question

Let \(\gamma\) be a closed curve in the complex plane (not necessarily differentiable) with parameter interval \([0,2 \pi]\), such that \(\gamma(t) \neq 0\) for every \(t \in[0,2 \pi]\). Choose \(\delta>0\) so that \(|\gamma(t)|>\delta\) for all \(t \in[0,2 \pi]\). If \(P_{1}\) and \(P_{2}\) are trigonometric polynomials such that \(\left|P_{j}(t)-\gamma(t)\right|<\delta / 4\) for all \(t \in[0,2 \pi]\) (their existence is assured by Theorem 8.15), prove that Ind \(\left(P_{1}\right)=\) Ind \(\left(P_{2}\right)\) by applying Exercise \(25 .\) Define this common value to be Ind \((\gamma)\). Prove that the statements of Exercises 24 and 25 hold without any differentiability assumption.

Short answer

Based on the given problem, we are required to prove that for any two trigonometric polynomials P_1(t) and P_2(t) with uniformly close distances to a closed curve γ(t), the winding numbers around the origin (indices) are equal while not assuming differentiability of the curve γ(t). The steps in the provided solution establish that: 1. Setting up the assumptions and definitions for the error between any curve α(t) and γ(t) and the related errors E(α_1) and E(α_2). 2. Using the Triangle Inequality and properties of indices to relate the indices of P_1 and P_2 through the error bound. 3. Proving that the indices of the two trigonometric polynomials are equal using the small error bound and defining the index of γ accordingly. 4. Removing differentiability assumptions by combining results from Exercise 25 and proving that the result is valid for non-differentiable cases as well. Hence, we have proven that the winding number (index) is robust to small perturbations in the curve, even for non-differentiable cases.

Step-by-step solution

Setup the assumptions and definitions

Let's begin by defining the required terms. First, let's define the winding number (also known as the index) of a closed curve around the origin: Ind\((\gamma) = \frac{1}{2 \pi} \oint_\gamma \frac{dz}{z}\) Now, let's define the error between any curve \(\alpha(t)\) and \(\gamma(t)\) as \(E(\alpha) = \sup_{t \in [0, 2 \pi]} |\alpha(t) - \gamma(t)|\). Using this, we can see that the curves \(\alpha_1(t) = \gamma(t) - P_1(t)\) and \(\alpha_2(t) = \gamma(t) - P_2(t)\) have errors with \(E(\alpha_1) < \delta/4\) and \(E(\alpha_2) < \delta/4\). Let's start by proving that the winding numbers for \(P_1\) and \(P_2\) are equal i.e., Ind\((P_1) = \) Ind\((P_2)\).

Use the Triangle Inequality and properties of indices to relate Ind\((P_1)\) and Ind\((P_2)\)

From a geometric perspective, we can obtain the following inequality based on the triangle formed by \(\alpha_1(t)\), \(\alpha_2(t)\), and \(\gamma(t)\): \(|\alpha_1(t) - \alpha_2(t)| \le E(\alpha_1) + E(\alpha_2)\). Since we have chosen \(\delta > 0\), we can bound \(|\alpha_1(t) - \alpha_2(t)|\) as follows: \(|\alpha_1(t) - \alpha_2(t)| < \delta/2\). Now, since the indices have the additive property where Ind\((\alpha_1+\alpha_2) = \text{Ind}(\alpha_1) + \text{Ind}(\alpha_2)\), we get: Ind\((\alpha_1 - \alpha_2) = \) Ind\((P_1 - P_2) = \) Ind\((\alpha_1) - \) Ind\((\alpha_2)\).

Use the small error bound to prove Ind\((P_1) = \) Ind\((P_2)\) and define Ind\((\gamma)\)

From the previous step, we have: Ind\((P_1-P_2) = \) Ind\((\gamma - P_2) - \) Ind\((\gamma - P_1) = \) Ind\((\alpha_1) - \) Ind\((\alpha_2)\). Since the error \(|\alpha_1(t)-\alpha_2(t)| < \delta/2\), by Exercise 25, we can guarantee that: Ind\((\alpha_1) = \) Ind\((\alpha_2)\). So, we have: Ind\((P_1 - P_2) = 0\). Thus, Ind\((P_1) = \) Ind\((P_2)\). Now we can define the index of \(\gamma\) as Ind\((\gamma) = \) Ind\((P_1)\), or equivalently, Ind\((\gamma) = \) Ind\((P_2)\).

Removing differentiability assumptions using the result from Exercise 25

From Exercise 25, we know that the winding number (index) around the origin is invariant under small perturbations (of size less than \(\delta/2\)) for non-differentiable curves. Combining this with the results we obtained in the previous steps, we can now conclude that the statements from Exercises 24 and 25 hold without any differentiability assumption. In summary, as long as we have two trigonometric polynomials \(P_1\) and \(P_2\) that approximate a closed curve \(\gamma\) such that the approximation errors are less than \(\delta/4\), the indices of \(\gamma\), \(P_1\), and \(P_2\) are equal, and this result holds without requiring the curve \(\gamma\) to be differentiable anywhere. This proves that the winding number (index) is robust against small perturbations in the curve, even for non-differentiable cases.

Key concepts

Complex Plane Curves

In mathematical analysis, a complex plane curve is a function that describes a path or outline in the complex plane. These curves are expressed in the form in which each point on the plane is a complex number corresponding to a particular value of t within a parameter interval, often [0, 2π] for closed curves.

Complex plane curves play a central role in complex analysis and topology. They are used to explore properties such as continuity, differentiability, and integration in the complex context. The winding number or index of a closed curve, which measures the total number of times the curve winds around a specific point, is a fundamental concept that relates to the behavior of these curves in the complex plane.

For example, when considering a closed curve in the complex plane that never passes through the origin ( for all t), it's essential to determine how many times the curve winds around the origin. This is where the notion of winding number becomes critical, providing a discrete quantity that captures the curve's topological essence without requiring the curve to be differentiable.

Trigonometric Polynomials

A trigonometric polynomial is a finite linear combination of sine and cosine functions, usually expressed as This sum defines a periodic function useful in approximating other functions on the interval [0, 2π] through techniques such as Fourier analysis.

In the context of the exercise, trigonometric polynomials are used to approximate the given complex plane curve , ensuring that the approximation error is within a specific bound. The theorem referenced in the exercise guarantees the existence of such polynomial approximations for any continuous function on a compact interval. These polynomials, , are critical in showing that the winding number of an approximate curve can be used to define the winding number of the original curve , illustrating the robustness of the winding number with respect to approximations made by trigonometric polynomials.

Differentiability in Mathematical Analysis

The concept of differentiability is a cornerstone of mathematical analysis. Differentiability refers to the ability to determine the derivative of a function at a point, which represents the function's instantaneous rate of change at that point. Functions that are differentiable at every point in an interval are referred to as smooth functions.

The exercise challenges the common prerequisite of differentiability for certain mathematical results. While many theorems in analysis assume the differentiability of functions, the notion that winding numbers are invariant under small non-differentiable perturbations suggests that some properties and theorems hold without this assumption. This robustness expands the applicability of such results to a broader class of functions and curves, which can have significant implications in complex analysis and other related fields.

Topology of Winding Number

In topology, the winding number is an integer that represents how a curve winds around a point in a specified direction. The winding number is a topological invariant, meaning it is preserved under continuous deformations of the curve as long as it doesn't pass through the point around which we are measuring.

Using the concept of trigonometric polynomial approximations, it can be demonstrated that the winding number is robust under approximate transformations. It is crucial in understanding and proving properties about complex functions and mappings without the strict necessity of differentiability. In other words, it provides a bridge between continuous and differential properties, allowing for the exploration of more general cases in the field of mathematical analysis.By establishing the equality of the indices of two approximate trigonometric polynomials, the exercise confirms that the winding number remains consistent for certain variations of non-differentiable curves, demonstrating the intrinsic topological nature of the winding number.