Problem 16 - Complex Analysis

Question

Let $$ D=\left\\{z \in \mathbb{C} ;-\frac{\pi}{2}<\operatorname{Re} z<\frac{\pi}{2}\right\\} $$ Show: (a) For \(f(z)=\sin z\) we have \(f(D)=\mathbb{C} \backslash\\{t \in \mathbb{R} ; \quad|t| \geq 1\\}\). (b) For \(f(z)=\tan z\) we have \(f(D)=C \backslash\\{t \mathrm{i} ; \quad t \in \mathbb{R}, t \geq 1\) or \(t \leq-1\\}\). The map tan : \(D \rightarrow f(D)\) is conformal, and its inverse is $$ g(z)=\frac{1}{2 \mathrm{i}} \log \frac{1+\mathrm{i} z}{1-\mathrm{i} z} $$

Step-by-Step Solution

Answer

(a) \(f(D) = \mathbb{C} \backslash \{t \in \mathbb{R}; |t| \geq 1\}\). (b) \(f(D) = \mathbb{C} \backslash \{t \mathrm{i}; |t| \geq 1 \}\).

Step 1: Understanding the Domain D

The domain \(D\) consists of all complex numbers \(z\) where the real part lies in the interval \(\left( -\frac{\pi}{2}, \frac{\pi}{2} \right)\). This interval corresponds to the open vertical strip in the complex plane.

Step 2: Analyzing the Sine Function on D

Consider \(f(z) = \sin z\). We need to find the image \(f(D)\). For real \(z\) within the interval \((-\frac{\pi}{2}, \frac{\pi}{2})\), \(\sin z\) achieves every value between \(-1\) and \(1\). For non-real \(z\), combinations of hyperbolic and trigonometric functions allow \(\sin z\) to achieve all complex values except those on the real axis with magnitude greater than 1.

Step 3: Result for Part (a)

The image \(f(D)\) for \(f(z) = \sin z\) covers the entire complex plane except real values with magnitude \(|t| \geq 1\). Hence, \(f(D) = \mathbb{C} \backslash \{t \in \mathbb{R} ; |t| \geq 1\}\).

Step 4: Considering the Tangent Function

For \(f(z) = \tan z\), the function is defined wherever \(z e \frac{\pi}{2} + k\pi\) for integer \(k\). Within the domain \(D\), \(\tan z\) maps to all complex numbers except purely imaginary numbers with magnitude greater than or equal to 1.

Step 5: Result for Part (b)

The image \(f(D)\) for \(f(z) = \tan z\) does not cover imaginary values \(t\mathrm{i}\) where \( \left| t \right| \geq 1\). It is denoted as \(\mathbb{C} \backslash \{t \mathrm{i} ; t \in \mathbb{R}, t \geq 1\) or \(t \leq -1\}\).

Step 6: Explaining the Conformality and Inverse

The function \( \tan z \) is conformal (angle-preserving) on \( D \) because it's holomorphic and has a non-zero derivative on this domain. The inverse, \(g(z) = \frac{1}{2\mathrm{i}} \log \frac{1 + \mathrm{i}z}{1 - \mathrm{i}z}\), is derived from the relationship between the tangent function and its complex logarithm.

Key Concepts

Conformal Mapping

Conformal mapping is a fascinating topic in complex analysis that deals with angle-preserving transformations between complex domains. These mappings are particularly powerful because they maintain the local angles and shapes of small structures. This is underpinned by the holomorphic nature of the functions involved.
In the context of the textbook problem, the function \(f(z) = \tan z\) is shown to be conformal on the domain \(D\). This is because the tangent function, being holomorphic, has a non-zero derivative throughout the domain. Conformal maps like this are vital in solving complex problems and modeling physical phenomena where the preservation of angles is crucial.
Such mappings open the door to practical applications in physics and engineering, especially in fluid dynamics and aerodynamics, where preserving the integrity of angles can be essential for predicting behavior and outcomes of systems.

Holomorphic Functions

Holomorphic functions are central to complex analysis. They are complex-valued functions that are differentiable at every point in their domain, which is a stronger condition than real differentiability due to the complexities of the complex plane. Differentiability in this context guarantees that the function's derivative is continuous, a property that lends stability and predictability to the function's behavior.
In the exercise, both \(\sin z\) and \(\tan z\) are holomorphic on their respective domains within \(D\), excluding the poles where \(\tan z\) is undefined. This property explains why concepts like conformal mapping arise, as holomorphic functions always have non-zero derivatives, which preserve angles locally.
The study of holomorphic functions extends into various branches of mathematics and physics, providing rich ground for theoretical and applied research.

Complex Functions

Complex functions, such as \(\sin z\) and \(\tan z\) in this exercise, are functions that accept complex numbers (written as \(z = x + yi\)) and return complex numbers. One of the most intriguing aspects of complex functions is how they blend real and imaginary components through operations like sum, product, and composition, resulting in behaviors and properties distinct from real-valued functions.
The domain of \(\sin z\) in our scenario, \(D\), allows a rich interplay of trigonometric and hyperbolic characteristics. Unlike their real counterparts, these functions don’t just swing back and forth within predictable bounds but instead cover expansive regions of the complex plane excluding specific paths or lines. This reality broadens the utility and flexibility of complex functions in modeling and simulations.
Understanding complex functions is key for students as it builds foundations for advanced applications extending beyond basic calculus.

Inverse Functions

Inverse functions 'reverse' the effect of a given function, recovering the original input. In complex analysis, finding the inverse of complex functions can be particularly challenging and insightful.
In the exercise, the inverse of \(\tan z\) within the domain \(D\) is given by:\\[g(z) = \frac{1}{2\mathrm{i}} \log \left(\frac{1 + \mathrm{i}z}{1 - \mathrm{i}z}\right)\] \ This expression reveals how inverse trigonometric functions often involve complex logarithms, due to their multi-valued nature and the branch cuts needed to define them uniquely.
Understanding and computing such inverses are important for solving equations involving complex functions, thus equipping learners with tools to tackle more intricate problems and unlock insights into the behavior of these mathematical entities.