Problem 7 - Complex Analysis

Question

In which domain \(D \subset \mathbb{C}\) defines the following series $$ \sum_{n=1}^{\infty} \frac{\sin (n z)}{2^{n}} $$ an analytic function? \((\) Answer: \(D=\\{z \in \mathbb{C} ; \quad|\operatorname{Im} z|<\log 2\\} .)\) Is there any domain in which the series $$ \sum_{n=1}^{\infty} \frac{\sin (n z)}{n^{2}} $$ defines an analytic function?

Step-by-Step Solution

Answer

The first series is analytic for \( |\operatorname{Im} z| < \log 2 \) and the second series is never analytic.

Step 1: Understand the First Series

The first series is \( \sum_{n=1}^{\infty} \frac{\sin (n z)}{2^n} \). We analyze the convergence by examining the term \( \frac{\sin (n z)}{2^n} \). For convergence, we need to assess the behavior of \( \sin(nz) \).

Step 2: Estimate the Sin Function

For complex \( z = x + iy \), we have \( \sin(nz) = \sin(nx + iny) = \sin(nx)\cosh(ny) + i\cos(nx)\sinh(ny) \). For the series to converge absolutely, we require that \( \cosh(ny) \) does not grow too fast.

Step 3: Convergence Condition for First Series

The term \( \cosh(ny) \approx \frac{e^{|y|n}}{2} \) as \( n \to \infty \). So, \( \left|\frac{\sin(nz)}{2^n}\right| \approx \left|\frac{e^{|y| n}}{2^{n+1}}\right| \). For convergence, we need \( \left|\frac{e^{|y| n}}{2^{n+1}}\right| < 1\) for all sufficiently large \( n \), i.e., \( |y| < \log 2 \). Thus, the domain is \( D = \{z \in \mathbb{C}; |\operatorname{Im} z| < \log 2\} \).

Step 4: Check Second Series for Analyticity

The second series is \( \sum_{n=1}^{\infty} \frac{\sin (n z)}{n^2} \). The numerator \( \sin(nz) \), using the same form as before, means \( \sin(nz) = \sin(nx)\cosh(ny) + i\cos(nx)\sinh(ny) \).

Step 5: Analyze Divergence in Second Series

For the series to be analytic, it must converge uniformly in some domain. \( \cosh(ny) \approx \frac{e^{|y|n}}{2} \) still grows too fast compared to \( n^2 \), as \( n^2 \) is polynomial growth and can't counteract the exponential growth of \( e^{|y|n} \) uniformly for any non-zero \( y \).

Step 6: Conclude Non-Existence of Such Domain

Thus, there is no domain in which the series \( \sum_{n=1}^{\infty} \frac{\sin (n z)}{n^2} \) converges uniformly and hence defines an analytic function, because the exponential growth in \( y \) cannot be handled by the \( n^2 \) denominator.

Key Concepts

Analytic Functions

Analytic functions are a special class of functions in complex analysis. A function is termed analytic over a domain if it is both differentiable and continuous in that domain.
This means that, at any given point within the specified domain, the function can be expressed as a power series.

• Being analytic implies having a derivative everywhere in the domain.
• Analytic functions are smooth and have well-behaved behavior in terms of continuity and differentiability.
• They can be locally expressed through a power series expansion.

Given a series, the determination of whether it corresponds to an analytic function typically revolves around the assessment of its convergence properties and behavior within a particular domain. In our specific exercise, examining functions composed of complex numbers requires assessing convergence with factors such as imaginary components.

Series Convergence

In complex analysis, series convergence is crucial in understanding whether a series defines an analytic function. For a series to converge, the sum of its terms must approach a finite limit as the number of terms grows infinitely large.
Two primary types of convergence are important:

• Absolute Convergence: This occurs if the series formed by taking the absolute value of each term also converges.
• Conditional Convergence: A series converges conditionally if it converges, but does not converge absolutely.

When dealing with series such as \[ \sum_{n=1}^{\infty} \frac{\sin (n z)}{2^{n}} \], the convergence is linked directly to evaluating terms like \(\frac{\sin(nz)}{2^n}\). For analytic properties to hold, convergence must be uniform across the specified domain. In our case, this involves determining constraints on the imaginary part of the complex number \(z\).

Complex Numbers

At the heart of complex analysis are complex numbers, which incorporate both real and imaginary components. A complex number \(z\) is typically expressed as \(z = x + iy\), where \(x\) is the real part and \(y\) is the imaginary part.
Analyzing functions composed of complex numbers involves appreciating how these components interact in expressions such as trigonometric functions.

• Real Part: The real component of the number, denoted as \(x\).
• Imaginary Part: The imaginary component, denoted as \(iy\).

For the series \(\sum_{n=1}^{\infty} \frac{\sin(nz)}{n^2}\), transforming \(\sin(nz)\) involves complex expressions, giving terms like \(\sin(nx)\cosh(ny) + i\cos(nx)\sinh(ny)\). Hence, analyzing convergence and domain involves handling both exponential and polynomial growth from this complex form.

Domain of Convergence

The domain of convergence is the set of values for which a series converges to a limit. For complex series, this domain is critical in identifying regions where the series defines analytic functions.
Finding this domain involves considering both the real and imaginary components of complex numbers, especially when series terms are exponential in nature.

• Imaginary Constraints: Often, series terms like \(\cosh(ny)\) factor in controlling convergence and lead to inequalities that identify this domain. For example, in \(\sum_{n=1}^{\infty} \frac{\sin(nz)}{2^{n}}\), convergence depends on \(|\operatorname{Im} z| < \log 2\).
• Analytic Implications: The domain is linked to where the series is well-behaved and analytical, necessitating a balance between exponential and polynomial growth properties.

In conclusion, determining this domain ensures the analyticity of series and helps avoid areas where exponential divergence might otherwise disrupt series properties.