Problem 10 - Complex Analysis

Question

Determine the maximum of \(|f|\) on \(\bar{E}:=\\{z \in \mathbb{C} ; \quad|z| \leq 1\\}\) for (a) \(f(z)=\exp \left(z^{2}\right)\), (b) \(f(z)=\frac{z+3}{z-3}\), (c) \(f(z)=z^{2}+z-1\), (d) \(f(z)=3-|z|^{2}\). In (d) the maximal modulus is attained in the (interior) point \(a=0\). Is there any contradiction to the maximum principle?

Step-by-Step Solution

Answer

The maximums are: (a) 1, (b) 2.236, (c) 1.414, (d) 3; (d) is not a contradiction.

Step 1: Define the Contour for Maximum Modulus

First, note that \(\bar{E}\) describes a closed disk with center at the origin and radius 1 in the complex plane. For each function \(f(z)\), we need to analyze \(|f(z)|\) on this contour.

Step 2: Analyze Function (a)

For \( f(z) = e^{z^2} \), the modulus is \(|f(z)| = |e^{z^2}| = e^{\text{Re}(z^2)}\). Since \(|z| \leq 1\), analyze the real part of \(z^2\). The maximum occurs at \(z = \pm i\) where \(\text{Re}(z^2) = 0\), giving maximum \(|f(z)| = 1\).

Step 3: Analyze Function (b) Using Boundary Values

For \( f(z) = \frac{z+3}{z-3} \), consider the modulus \(\left|\frac{z+3}{z-3}\right|\). Calculating at \( z=1 \), yields a maximum value of \(2.236\) approximately.

Step 4: Analyze Function (c) Direct Evaluation

For \( f(z) = z^2 + z - 1 \) with \(|z| \leq 1\), evaluate \(f(z)\) at the boundary, such as \(z = 1\), \(z = -1\), and \(z = i\). The maximum value of \(|f(z)|\) is approximately \(1.414\).

Step 5: Evaluate Function (d) at Specific Points

For \( f(z) = 3 - |z|^2 \), note that \(3\) is constant minus a nonnegative quantity. The maximum occurs at \(z = 0\), where \(3 - 0 = 3\). Since \(f\) achieves its maximum inside the disk, this is not a contradiction to the maximum principle as \(f(z)\) is not analytic (depends on \(|z|^2\)).

Key Concepts

Complex Functions

Complex functions are mathematical expressions involving complex numbers. These numbers have both a real and an imaginary part, expressed in the form \( z = x + yi \), where \( x \) and \( y \) are real numbers, and \( i \) is the imaginary unit with the property \( i^2 = -1 \). Complex functions like the ones described in the exercise can have various forms, such as exponential (e.g., \( f(z) = e^{z^2} \)) or rational (e.g., \( f(z) = \frac{z+3}{z-3} \)).
These functions take inputs from the complex plane and yield complex outputs. Understanding the behavior and limitations of complex functions is crucial for solving problems involving the complex plane, where analytic properties can greatly influence the function's characteristics.

Analyticity

Analyticity refers to a property of complex functions that are differentiable at every point in some region of the complex plane. When a function is analytic, it has derivatives of all orders, and these functions are often referred to as holomorphic functions.
To be analytic, a function must satisfy specific conditions, one of which involves the famous Cauchy-Riemann equations. These equations ensure that the real and imaginary parts of the function are interlinked, allowing for a smooth variation across the complex plane. For instance, in the original problem, function (a) \( f(z) = e^{z^2} \) is analytic within the discussed circle \( \bar{E} \), enabling us to leverage the maximum modulus principle, a critical concept in the evaluation of maximum values within a bounded region.

Complex Plane

The complex plane is a two-dimensional plane used to visualize complex numbers. The horizontal axis represents the real part, while the vertical axis stands for the imaginary part.

• Every point on this plane denotes a complex number.
• The modulus of a complex number \( z = x + yi \) is calculated as \( |z| = \sqrt{x^2 + y^2} \), representing its distance from the origin.
• The unit circle, centered at the origin with radius 1, is an essential concept often used in problems involving maximum modulus.

In the given exercise, all calculations are analyzed within or on a closed disk, \( \bar{E} \), which represents this unit circle and its interior. Understanding how functions behave on the complex plane, especially within specified boundaries like \( \bar{E} \), is vital for applying theories like maximum modulus and analyzing function behavior.

Maximum Modulus Theorem

The Maximum Modulus Theorem is a principal theorem in complex analysis. It asserts that if a function is analytic and non-constant within a closed and bounded region, then the maximum value of its modulus occurs on the boundary of that region. This theorem helps identify the point within a specified area where the maximum occurs.
In our exercise:

• Function (a) \( f(z) = e^{z^2} \) demonstrates this by attaining its maximum modulus on the boundary.
• Function (d) \( f(z) = 3 - |z|^2 \) is not analytic as it depends directly on \( |z|^2 \), demonstrating why the theorem is not applicable.

Analyzing these scenarios helps in understanding how analytic properties and the maximum modulus theorem interact with complex functions. This knowledge is essential, especially in fields where the properties of analytic functions are leveraged, such as engineering and physics.