Problem 10 - Complex Analysis

Question

A somewhat more direct proof of the generalized CAUCHY integral formula (Theorem II.3.4) is obtained with the following Lemma: Let \(\alpha:[a, b] \rightarrow \mathbb{C}\) be a piecewise smooth curve and let \(\varphi\) : Image \(\alpha \rightarrow \mathbb{C}\) be continuous. For \(z \in D:=\mathbb{C} \backslash\) Image \(\alpha\) and \(m \in \mathbb{N}\) let $$ F_{m}(z):=\frac{1}{2 \pi \mathrm{i}} \int_{\alpha} \frac{\varphi(\zeta)}{(\zeta-z)^{m}} d \zeta $$ Then \(F_{m}\) is analytic in \(D\) and for all \(z \in D\) $$ F_{m}^{\prime}(z)=m F_{m+1}(z) $$ Prove this by direct estimate (without using the LEIBNIZ rule).

Step-by-Step Solution

Answer

Direct differentiation shows \( F_m^{\prime}(z) = m F_{m+1}(z) \).

Step 1: Define Function Properties

Begin by acknowledging the properties of the analytic function. The function \( F_m(z) \) is defined as \( \frac{1}{2 \pi \mathrm{i}} \int_{\alpha} \frac{\varphi(\zeta)}{(\zeta-z)^{m}} d\zeta \). Here, \( \alpha \) is a piecewise smooth curve and \( \varphi \) is a continuous function on the image of \( \alpha \). The domain \( D \) is \( \mathbb{C} \setminus \text{Image } \alpha \). Since \( z \in D \), \( F_m \) is analytic if the integrand is analytic in \( D \), which it is due to continuity and smoothness conditions.

Step 2: Derivative of the Integral Expression

Consider \( F_m^{\prime}(z) \), which depends on the derivative under the integral sign. We cannot directly use LEIBNIZ rule here, so we do it manually by differentiating the expression:\(F_m^{\prime}(z) = \frac{d}{dz}\left( \frac{1}{2 \pi \mathrm{i}} \int_{\alpha} \frac{\varphi(\zeta)}{(\zeta-z)^{m}} d\zeta \right).\)Use differentiation under the integral (valid due to smoothness assumptions) leading to:\(F_m^{\prime}(z) = \frac{1}{2 \pi \mathrm{i}} \int_{\alpha} \frac{\partial}{\partial z} \left( \frac{\varphi(\zeta)}{(\zeta-z)^{m}} \right) d\zeta.\)

Step 3: Compute the Derivative of the Integrand

Differentiate the integrand \( \frac{\varphi(\zeta)}{(\zeta-z)^{m}} \) with respect to \( z \):\(\frac{\partial}{\partial z} \left( \frac{\varphi(\zeta)}{(\zeta-z)^{m}} \right) = m \frac{\varphi(\zeta)}{(\zeta-z)^{m+1}}.\)Substitute this result back into the integral:

Step 4: Substitute and Simplify

Substituting the derived expression into the integral for \( F_m^{\prime}(z) \):\(F_m^{\prime}(z) = \frac{1}{2 \pi \mathrm{i}} \int_{\alpha} m \frac{\varphi(\zeta)}{(\zeta-z)^{m+1}} d\zeta.\)This can be rewritten as:\(F_m^{\prime}(z) = m \frac{1}{2 \pi \mathrm{i}} \int_{\alpha} \frac{\varphi(\zeta)}{(\zeta-z)^{m+1}} d\zeta = m F_{m+1}(z).\)Thus, we have proven that \( F_m^{\prime}(z) = m F_{m+1}(z) \).

Key Concepts

Analytic Function

Analytic functions are fundamental in complex analysis. These functions are complex-valued and differentiable at every point in their domain. When a function is analytic on a domain, it means it can be represented as a power series within that domain. In simpler terms,

• Analytic functions have derivatives of all orders, meaning you can keep differentiating them, and they will behave well.
• To be analytic in a domain, a function must have no breaks, jumps, or undefined points in that region.

In the context of the Cauchy Integral Formula, when we mention that a function like \[ F_m(z) \] is analytic in a domain \( D \), it means that \( F_m(z) \) follows these properties throughout \( D \), which in our exercise is defined as \( \mathbb{C} \setminus \text{Image } \alpha \). The core idea is that if you stay within this domain, the function remains well-behaved and infinitely differentiable. It is these properties of analytic functions that make them predictable and very powerful in the field of complex analysis.

Piecewise Smooth Curve

In complex analysis, the idea of a piecewise smooth curve is pivotal when dealing with integrals. A piecewise smooth curve is essentially a curve that may not be entirely smooth but can be broken down into a finite number of smooth segments. Here's what it generally involves:

• The curve is continuous, meaning there are no jumps or gaps.
• It consists of segments where each segment is smooth — it has a well-behaved derivative.
• Each point where segments meet, known as "corners," is finite, ensuring the overall curve remains manageable.

In our exercise, the curve \( \alpha \) is piecewise smooth and it forms the path of integration. Why does this matter? Because the smoothness conditions ensure that the complex integral, as defined, behaves predictably. It guarantees that the function being integrated, \( \varphi(\zeta) \), impacts the integral only in ways we can accurately assess, leading to a consistent result. This characteristic is vital when proving properties like the analytic nature of \( F_m(z) \).

Derivative Under the Integral Sign

Finding the derivative under an integral sign is a technique used when a variable appears within an integral. Doing this correctly is crucial when differentiating an integral, as it relates to the Leibniz rule. However, in contexts like our exercise, we find the derivative manually. Here's how it works:

• We focus on differentiating an integral expression directly by differentiating the integrand first.
• In our case, we differentiate \( \frac{\varphi(\zeta)}{(\zeta-z)^{m}} \) with respect to \( z \), yielding \( m \frac{\varphi(\zeta)}{(\zeta-z)^{m+1}} \).
• After differentiation, this new expression is substituted back into the integral.

This leads us to calculate \( F_m^{\prime}(z) \) without using the Leibniz rule directly. The idea here is that under smoothness assumptions about the curve and the integrand, this technique is valid and helps in securing the differentiability of the function \( F_m(z) \) around the domain \( D \).

Complex Domain

The term complex domain refers to a subset of the complex plane where complex functions are analyzed. In complex analysis, the complex plane serves as the platform where all points are labeled with complex numbers. A complex domain, like the one in our exercise, \( D = \mathbb{C} \setminus \text{Image } \alpha \), is critical for:

• Determining where the function behaves analytically. In our context, \( F_m(z) \) is analytic within this domain.
• Understanding the influence of the contours or curves on the interaction of the function within and outside these domains.
• Ensuring that the path of integration does not cross into regions where the function might be undefined or non-analytic.

The concept of a complex domain is crucial for the accurate application of the Cauchy Integral Formula and proves essential when verifying properties of functions like \( F_m^{\prime}(z) \). It serves as a defined space where mathematicians can work with complex functions, ensuring expected properties like differentiability remain valid throughout that space.