Problem 10 - Complex numbers from A to...Z

Question

$$ \begin{aligned} &\text { Find }\\\ &\min _{z \in \mathbb{C} \backslash \mathbb{R}} \frac{\operatorname{Im} z^{5}}{\operatorname{Im}^{5} z} \end{aligned} $$ and the values of \(z\) for which the minimum is reached.

Step-by-Step Solution

Answer

Answer: The minimum value of the expression is \(\frac{1}{18}\), and it is reached for the values of \(z = re^{i\theta}\), where \(r^4\sin^4\theta = 18\) and \(\theta = \frac{\pi}{10} + \frac{2n\pi}{5}\) for \(n=0,1,2,3,4\).

Step 1: Rewrite the expression in terms of polar coordinates

Since we're dealing with complex numbers, it's often useful to rewrite the expression in terms of polar coordinates. Let \(z = re^{i\theta}\), where \(r > 0\). Then, $$\operatorname{Im}(z) = r\sin\theta$$ and $$\operatorname{Im}(z^5) = r^5\sin(5\theta).$$ Now we can rewrite the given expression as $$ \frac{\operatorname{Im}(z^5)}{\operatorname{Im}^5(z)} = \frac{r^5\sin(5\theta)}{(r\sin\theta)^5} = \frac{1}{r^4\sin^4\theta}\sin(5\theta) $$

Step 2: Apply the AM-GM inequality

To find the minimum value of the expression, we'll use the AM-GM inequality. The AM-GM inequality states that for non-negative real numbers \(x_1, x_2, ..., x_n\), their arithmetic mean is greater than or equal to their geometric mean: $$ \frac{x_1+x_2+\cdots+x_n}{n} \geq \sqrt[n]{x_1x_2\cdots x_n} $$ Applying the AM-GM inequality for the numbers \(\{1,1,1,r^4\sin^4\theta\}\), we have $$ \frac{1+1+1+r^4\sin^4\theta}{4} \geq (1\cdot 1\cdot 1\cdot r^4\sin^4\theta)^{1/4} $$ Hence, \(r^4\sin^4\theta \geq 3^3-3^2 = 27-9 = 18\). Now we have a lower bound for \(r^4\sin^4\theta\), and we can find the minimum value of the expression: $$ \frac{1}{r^4\sin^4\theta}\sin(5\theta) \geq \frac{1}{18}\sin(5\theta) $$

Step 3: Determine the minimum value and corresponding values of \(z\)

The minimum value of the expression is reached when \(\sin(5\theta) = 1\), so the minimum value is \(\frac{1}{18}\). The values of \(z\) for which this minimum is reached are those with \(r^4\sin^4\theta = 18\) and \(\sin(5\theta) = 1.\) From our above computation, this happens when \(5\theta = \frac{\pi}{2} + 2n\pi\) for \(n\in\mathbb{Z}\). Therefore, we have \(\theta = \frac{\pi}{10} + \frac{2n\pi}{5}\), where \(n=0,1,2,3,4.\) Finally, we have found that the minimum value of the expression is \(\frac{1}{18}\), and it is reached for the values of \(z = re^{i\theta}\), where \(r^4\sin^4\theta = 18\) and \(\theta = \frac{\pi}{10} + \frac{2n\pi}{5}\) for \(n=0,1,2,3,4\).

Key Concepts

Polar Coordinates in Complex Numbers

Understanding the relationship between complex numbers and polar coordinates is foundational when dealing with problems involving trigonometric properties or interpretations of complex functions. A complex number, typically written as \( z = x+iy \), can be alternatively represented in polar form as \( z = re^{i\theta} \), where \( r \) is the magnitude (modulus) of \( z \) and \( \theta \) is the argument, or angle, that \( z \) makes with the positive real axis.

In polar coordinates, the real part of \( z \) is \( r\cos\theta \) and the imaginary part is \( r\sin\theta \). This representation becomes especially useful when raising complex numbers to powers or extracting roots. The polar form simplifies multiplication and division and is key in solving complex number equations that would otherwise be tedious in Cartesian form. For the given problem, using the property that \( \operatorname{Im}(z) = r\sin\theta \) and \( \operatorname{Im}(z^5) = r^5\sin(5\theta) \), the expressiveness of polar coordinates is brought to light: it simplifies the equation and reveals underlying periodic and multiplicative properties that are essential to finding the solution.

AM-GM Inequality Application

The Arithmetic Mean-Geometric Mean (AM-GM) inequality is a fundamental principle in mathematics, frequently leveraged to find minimum or maximum values in optimization problems. It states that for a set of non-negative real numbers, the arithmetic mean is always greater than or equal to the geometric mean, with equality only when all the numbers are equal.

The equation \( \frac{1+1+1+r^4\sin^4\theta}{4} \geq (1\cdot 1\cdot 1\cdot r^4\sin^4\theta)^{1/4} \) emerging from the AM-GM inequality sets a lower bound for \( r^4\sin^4\theta \), enabling us to find the minimum value for the complex number problem at hand. This application of the AM-GM inequality simplifies the problem by reducing the region of possible solutions to the inequality \( r^4\sin^4\theta \geq 18 \), thereby paving the path to finalizing the minimum value of the given expression.

Solving Complex Number Equations

When solving equations involving complex numbers, it's often useful to convert the complex number into its polar form as it simplifies many operations. In the given minimum value problem, after utilizing polar coordinates and applying the AM-GM inequality, it becomes clear that the task is to find the values of \( z \) that satisfy the equation \( \sin(5\theta) = 1 \), as well as the condition \( r^4\sin^4\theta = 18 \).

This requires knowledge of sine function properties, notably its periodicity and the specific angles for which it can take values of 1. By recognizing that \( \sin(5\theta) = 1 \) at \( \theta = \frac{\pi}{2n} \) for integers \( n \), we narrow down the potential angles that satisfy the equation, leading to the specific \( \theta \) values that give us the minimum. In the broader context of solving complex number equations, this step is analogous to finding the zeros of a function or the roots of a polynomial, with the added richness of complex analysis.