Question

Let \(\gamma\) be a continuously differentiable closed curve in the complex plane, with parameter interval \([a, b]\), and assume that \(\gamma(t) \neq 0\) for every \(t \in[a, b]\). Define the index of \(\gamma\) to be $$ \operatorname{Ind}(\gamma)=\frac{1}{2 m i} \int_{e}^{t} \frac{\gamma(t)}{\gamma(t)} d t $$ Prove that Ind \((\gamma)\) is always an integer. Hint: There exists \(\varphi\) on \([a, b]\) with \(\varphi^{\prime}=\gamma^{\prime} / \gamma, \varphi(a)=0\). Hence \(\gamma \exp (-\varphi)\) is constant. Since \(\gamma(a)=\gamma(b)\) it follows that \(\exp \varphi(b)=\exp \varphi(a)=1\). Note that \(\varphi(b)=2 \pi i\) Ind \((\gamma)\). Compute Ind \((\gamma)\) when \(\gamma(t)=e^{i * \prime}, a=0, b=2 \pi\) Explain why Ind \((\gamma)\) is often called the winding number of \(\gamma\) around \(0 .\)

Short answer

Based on the given step-by-step solution, answer the following: Q: Prove that the index of a continuously differentiable closed curve in the complex plane is always an integer. A: To prove that, we first define a function \(\varphi\) on the interval \([a, b]\), where \(\varphi^{\prime}(t) = \frac{\gamma^{\prime}(t)}{\gamma(t)}\), and \(\varphi(a) = 0\). Next, we show that the function \(\gamma(t)e^{-\varphi(t)}\) is constant by calculating its derivative, which is equal to zero. Next, using the fact that the curve is closed, we show that \(e^{\varphi(a)} = e^{\varphi(b)} = 1\). Finally, we use the property that \(\varphi(b) = 2\pi i\) Ind(\(\gamma\)), to conclude that the index, Ind(\(\gamma\)), must be an integer since it is the factor inside the exponential.

Step-by-step solution

Define \(\varphi\) with the given properties

Follow the hint and consider a function \(\varphi\) on \([a, b]\) where \(\varphi^{\prime}(t) = \frac{\gamma^{\prime}(t)}{\gamma(t)}\) and \(\varphi(a) = 0\). Such a function exists, as the curve is differentiable, and \(\gamma(t) \neq 0\) for all \(t \in[a, b]\).

Show that \(\gamma(t)e^{-\varphi(t)}\) is constant

To show that the function \(\gamma(t)e^{-\varphi(t)}\) is constant, we can compute its derivative and show that it's equal to zero for all \(t \in [a, b]\): $$ \frac{d}{dt}\left(\gamma(t)e^{-\varphi(t)}\right) = \frac{d\gamma(t)}{dt}e^{-\varphi(t)} - \gamma(t)e^{-\varphi(t)}\frac{d\varphi(t)}{dt} = \gamma^{\prime}(t)e^{-\varphi(t)} - \gamma(t)e^{-\varphi(t)}\frac{\gamma^{\prime}(t)}{\gamma(t)} = 0 $$ Since the derivative is zero, \(\gamma(t)e^{-\varphi(t)}\) is constant.

Use the fact that \(\gamma(a)=\gamma(b)\) to show \(e^{\varphi(b)}=e^{\varphi(a)}=1\)

The curve is closed, meaning \(\gamma(a) = \gamma(b)\). Since \(\gamma(t)e^{-\varphi(t)}\) is constant, we have \(\gamma(a)e^{-\varphi(a)} = \gamma(b)e^{-\varphi(b)}\). But we know that \(\varphi(a) = 0\), so \(e^{-\varphi(a)} = 1\), and therefore \(e^{\varphi(a)} = 1\). Since \(\gamma(a) = \gamma(b)\), it follows that \(e^{\varphi(b)} = e^{\varphi(a)} = 1\).

Conclude that the index Ind(\(\gamma\)) is an integer

From the hint, we know that \(\varphi(b) = 2\pi i\) Ind(\(\gamma\)). We have shown that \(e^{\varphi(b)} = 1\). When \(1 = e^{2\pi i \text{Ind}(\gamma)}\), the factor inside the exponential must be an integer multiple of \(2\pi i\), meaning Ind(\(\gamma\)) must be an integer.

Compute Ind(\(\gamma\)) for \(\gamma(t)=e^{it}, a=0, b=2\pi\)

When \(\gamma(t) = e^{it}\), \(a = 0\), and \(b = 2\pi\), we have: $$ \text{Ind}(\gamma) = \frac{1}{2\pi i} \int_{0}^{2\pi} \frac{ie^{it}}{e^{it}} dt = \frac{1}{2\pi i} \int_{0}^{2\pi} i\, dt. $$ Integrating with respect to \(t\), we get: $$ \text{Ind}(\gamma) = \frac{1}{2\pi i}\left[ti\right]_0^{2\pi} = \frac{1}{2\pi i} (2\pi i - 0) = 1. $$

Explain why Ind(\(\gamma\)) is called the winding number around \(0\)

In this example, Ind(\(\gamma\)) is equal to \(1\), meaning the curve \(\gamma(t) = e^{it}\) winds once around the point \(0\) in the complex plane. In general, Ind(\(\gamma\)) represents the number of times the curve winds around the point \(0\), either in the positive direction (counterclockwise) if the index is positive, or in the negative direction (clockwise) if the index is negative. Thus, it is called the winding number of the curve around point \(0\).

Key concepts

Winding Number

The winding number is a fundamental concept in complex analysis and topology that describes the number of times a curve wraps around a particular point in the complex plane. It is an integer value that can be positive, negative, or zero. In our discussion, we've used the term 'index' synonymously with the winding number.

The winding number is significant in complex analysis, as it helps mathematicians understand the behavior of complex functions. For instance, if a curve wraps around a point multiple times in a counterclockwise direction, the winding number is a positive integer. Conversely, if it wraps around clockwise, the winding number is a negative integer. A winding number of zero means the curve does not encircle the point at all.

This concept is particularly useful when analyzing how functions behave near singularities. In the context of the given exercise, the winding number of the curve \(\gamma\) around the origin is calculated. It's a way to quantify the 'twisting' or 'coiling' of \(\gamma\) around a reference point, which in this case, is zero.

Continuously Differentiable Closed Curve

A continuously differentiable closed curve, often found in complex analysis, is a smooth path in the complex plane that returns to its starting point without any sharp edges or corners. In other words, its derivative exists at every point, and it forms a loop. For such a curve, defined on a parameter interval \( [a, b] \), the property that \(\gamma(a) = \gamma(b)\) is essential.

This characteristic of being 'closed' makes it possible to discuss concepts like the winding number, because the notion of winding around a point becomes meaningful only when the path completes a loop. If a curve isn't closed, it wouldn't fully encircle any point, and so the winding number wouldn't be well-defined.

In our exercise, \(\gamma\) being continuously differentiable is also critical for the integrand \(\gamma'(t)/\gamma(t)\) to make sense, assuring us that the calculations involved in determining the index are possible. This smoothness ensures there are no discontinuities or infinite gradients on the curve, which aids in performing integration and other analytical processes without ambiguity.

Complex Analysis

Complex analysis is a branch of mathematics focusing on functions of a complex variable. It encompasses a wide array of subjects including holomorphic functions, contour integration, and conformal mappings, among others. Central to complex analysis is the study of complex differentiable functions, which are often referred to as analytic functions.

In the context of the provided exercise, complex analysis offers tools like contour integration to compute the winding number of curves. As we integrate a function over a path in the complex plane, we are essentially summing up the effects of the function along that path, which is crucial in applications ranging from evaluating integrals to solving boundary-value problems in physics.

An important theorem in complex analysis used in the solution of the exercise is Cauchy's Integral Theorem, which states that under certain conditions, the integral of a complex function along a closed curve is zero. This supports step 2 where we show that \(\gamma(t)e^{-\varphi(t)}\) is constant, since its derivative is zero. Complex analysis provides the theoretical foundation necessary to understand why the index of a curve (the winding number) must be an integer.