Problem 14 - Complex Analysis

Question

Let \(D=\\{z \in \mathbb{C} ; \quad-\pi<\operatorname{Im} z<\pi, 0<\operatorname{Re} z

Step-by-Step Solution

Answer

f is a conformal map from D to \(D' = \{ w \in \mathbb{C} : 0 < |w| < e^b \} \).

Step 1: Understand the Definition of Domain D

The domain \(D\) is given as \(\{z \in \mathbb{C} ; \ -\pi<\operatorname{Im} z<\pi, \ 0<\operatorname{Re} z

Step 2: Consider the Mapping f = exp

The function \(f(z) = \exp(z)\) maps the complex plane to itself through the exponential function. The exponential function takes a complex number \(z = x + iy\) to \(e^x(\cos y + i\sin y)\), affecting both magnitude and angle (or winding).

Step 3: Map the Domain using the Exponential Function

The exponential function \(f(z) = e^z\) maps vertical strips to annular or circular regions. Specifically, for a vertical strip like \(D\), \(e^z\) will wrap the strip around the origin while stretching according to the real part. The line segment \(\operatorname{Re} z = x\) maps to the curve \(r = e^x\) in the complex plane.

Step 4: Determine the Image of D under f

The region \(D\) defined by the strip \(0 < \operatorname{Re} z < b\) and \(-\pi < \operatorname{Im} z < \pi\) is transformed by \(f(z) = e^z\). The boundary \(\operatorname{Im} z = -\pi\) to \(\pi\) covers all directions spanning a circle (full range of an angle). Therefore, \(D' = \{ w \in \mathbb{C} : 0 < |w| < e^b \} \) is an annular region excluding the origin, with the radius determined by \(\operatorname{Re} z\).

Step 5: Prove the Mapping is Conformal

A conformal map preserves angles and is analytic with a non-zero derivative. The exponential function is entire (analytic over all \(\mathbb{C}\)) and its derivative, \(f'(z) = e^z\), is never zero since \(e^z eq 0\) for any complex \( z\). Thus, \(f\) is conformal on \(D\).

Key Concepts

Complex Exponential Function

The **Complex Exponential Function** plays a vital role in deeply understanding the behavior of complex functions across the complex plane. At its core, the exponential function, denoted as \( f(z) = \exp(z) \), takes a complex number \( z = x + iy \) and transforms it into a point \( e^x (\cos y + i \sin y) \). This is essentially represented by Euler's formula which elegantly combines exponential growth with rotations on the complex plane.

**Key properties of the complex exponential function include:**

• **Magnitude Modification**: The real part \( x \) influences the magnitude. Specifically, \( e^x \) represents the stretching factor that scales the resultant complex number's magnitude.
• **Angle Changes**: The imaginary part \( y \) dictates rotational movement as it essentially modulates the angle. This periodic nature results in the unique wrapping effect of vertical strips when applying \( f(z) \).
• **Periodic Nature**: Due to the inherent periodicity of sine and cosine functions, \( e^{z + 2\pi i} = e^z \) holds, demonstrating that the function repeats for every increment of \( 2\pi \) in the imaginary part.

These fundamental characteristics are leveraged in complex analysis to model phenomena like conformal mappings, where transformations must preserve the structure and angles of the mapped domain.

Analytic Function

An **Analytic Function** is a type of complex function crucial for understanding mappings in complex analysis. A function \( f(z) \) is deemed analytic at a point if it is differentiable at every point in a neighborhood around that point. More broadly, functions exhibit analyticity within a domain if such conditions hold at every point within that domain.

This differentiability of \( f(z) = \exp(z) \) is defined by the function's series representation and its derivative's consistency across the complex plane. For \( \exp(z) \), it’s important to note:

• **Everywhere differentiable**: Unlike real functions which have isolated points of non-differentiability, \( \exp(z) \) is differentiable everywhere in \( \mathbb{C} \) – a feature described as being entire.
• **Derivative**: The derivative \( f'(z) = e^z \) never becomes zero, ensuring the function’s non-zero slope preserving angles.
• **Conformal property**: By being everywhere analytic and non-zero derivative, \( \exp(z) \) is able to maintain structures within the domain, a crucial feature of conformal mappings.

In essence, the analytic nature of \( \exp(z) \) ensures that the transformation of domains maintains essential geometric properties, whilst facilitating conformal mappings that are angle-preserving.

Complex Domain

Understanding the **Complex Domain** is fundamental when working with conformal mappings and functions like the complex exponential. The complex domain defines the set of values at which a function is analyzed and manipulated, providing the canvas for transformations.

For the given problem, the domain \( D = \{ z \in \mathbb{C} : -\pi < \operatorname{Im} z < \pi, 0 < \operatorname{Re} z < b \} \) represents a vertical strip on the complex plane. This forms the initial area under consideration for set transformations using the exponential function. Let’s explore its characteristics:

• **Vertical Strip**: Defined by fixed imaginary components while the real component spans from 0 to \( b \), allowing us to visualize its structure as infinitely tall and "open-ended horizontal wall."
• **Imaginary Boundaries**: The boundary values \( -\pi \) and \( \pi \) help in understanding the rotational behavior after transformation with \( e^z \), wrapping the strip around the origin.
• **Transformation Result**: Applying \( f(z) = e^z \) transitions the strip into an annular region \( D' \) in the form \( \{ w \in \mathbb{C} : 0 < |w| < e^b \} \). This transformation is key in identifying the new conformal image where the function’s radically transformed outputs circulate in circular and annular patterns surrounding the origin.

Overall, comprehending the initial complex domain and the resulting transformed domain provides insight into how exponential functions navigate and reshape complex planes.

Previous

Problem 14

Next

Problem 15

Other exercises in this chapter

Problem 13

Let \(n \geq 2\) be a natural number. There is no function \(f: \mathbb{C}^{*} \rightarrow \mathbb{C}^{*}\) with the two properties (a) \(\quad f(z w)=f(z) f(w)

View solution

Problem 14

Let \(n \geq 2\) be a natural number. There is no continuous function \(q_{n}: \mathbb{C} \rightarrow \mathbb{C}\) such that $$ \left(q_{n}(z)\right)^{n}=z \tex

View solution

Problem 15

(a) Consider the map $$ f: \mathbb{C}^{*} \longrightarrow \mathbb{C} \text { with } f(z)=1 / \bar{z} $$ Give a geometrical construction (with ruler and compass)

View solution

Problem 15

The Joukowski function -named after the Russian mathematician N.J. JOUKOWSKI \((1847-1921)-\) $$ f: C^{*} \longrightarrow \mathbb{C}, \quad z \mapsto \frac{1}{2

View solution

Key Concepts

Complex Exponential Function

The **Complex Exponential Function** plays a vital role in deeply understanding the behavior of complex functions across the complex plane. At its core, the exponential function, denoted as \( f(z) = \exp(z) \), takes a complex number \( z = x + iy \) and transforms it into a point \( e^x (\cos y + i \sin y) \). This is essentially represented by Euler's formula which elegantly combines exponential growth with rotations on the complex plane.

**Key properties of the complex exponential function include:**

• **Magnitude Modification**: The real part \( x \) influences the magnitude. Specifically, \( e^x \) represents the stretching factor that scales the resultant complex number's magnitude.
• **Angle Changes**: The imaginary part \( y \) dictates rotational movement as it essentially modulates the angle. This periodic nature results in the unique wrapping effect of vertical strips when applying \( f(z) \).
• **Periodic Nature**: Due to the inherent periodicity of sine and cosine functions, \( e^{z + 2\pi i} = e^z \) holds, demonstrating that the function repeats for every increment of \( 2\pi \) in the imaginary part.

These fundamental characteristics are leveraged in complex analysis to model phenomena like conformal mappings, where transformations must preserve the structure and angles of the mapped domain.

Analytic Function

An **Analytic Function** is a type of complex function crucial for understanding mappings in complex analysis. A function \( f(z) \) is deemed analytic at a point if it is differentiable at every point in a neighborhood around that point. More broadly, functions exhibit analyticity within a domain if such conditions hold at every point within that domain.

This differentiability of \( f(z) = \exp(z) \) is defined by the function's series representation and its derivative's consistency across the complex plane. For \( \exp(z) \), it’s important to note:

• **Everywhere differentiable**: Unlike real functions which have isolated points of non-differentiability, \( \exp(z) \) is differentiable everywhere in \( \mathbb{C} \) – a feature described as being entire.
• **Derivative**: The derivative \( f'(z) = e^z \) never becomes zero, ensuring the function’s non-zero slope preserving angles.
• **Conformal property**: By being everywhere analytic and non-zero derivative, \( \exp(z) \) is able to maintain structures within the domain, a crucial feature of conformal mappings.

In essence, the analytic nature of \( \exp(z) \) ensures that the transformation of domains maintains essential geometric properties, whilst facilitating conformal mappings that are angle-preserving.

Complex Domain

Understanding the **Complex Domain** is fundamental when working with conformal mappings and functions like the complex exponential. The complex domain defines the set of values at which a function is analyzed and manipulated, providing the canvas for transformations.

For the given problem, the domain \( D = \{ z \in \mathbb{C} : -\pi < \operatorname{Im} z < \pi, 0 < \operatorname{Re} z < b \} \) represents a vertical strip on the complex plane. This forms the initial area under consideration for set transformations using the exponential function. Let’s explore its characteristics:

• **Vertical Strip**: Defined by fixed imaginary components while the real component spans from 0 to \( b \), allowing us to visualize its structure as infinitely tall and "open-ended horizontal wall."
• **Imaginary Boundaries**: The boundary values \( -\pi \) and \( \pi \) help in understanding the rotational behavior after transformation with \( e^z \), wrapping the strip around the origin.
• **Transformation Result**: Applying \( f(z) = e^z \) transitions the strip into an annular region \( D' \) in the form \( \{ w \in \mathbb{C} : 0 < |w| < e^b \} \). This transformation is key in identifying the new conformal image where the function’s radically transformed outputs circulate in circular and annular patterns surrounding the origin.

Overall, comprehending the initial complex domain and the resulting transformed domain provides insight into how exponential functions navigate and reshape complex planes.