Problem 7 - Complex Analysis

Question

For all \(z=x+\mathrm{i} y \in \mathbb{C}\) one has: (a) $$ \overline{\exp (z)}=\exp (\bar{z}), \quad \overline{\sin (z)}=\sin (\bar{z}), \quad \overline{\cos (z)}=\cos (\bar{z}) $$ (b) $$ \begin{aligned} &\cos z=\cos (x+\mathrm{i} y)=\cos x \cosh y-\mathrm{i} \sin x \sinh y \\ &\sin z=\sin (x+\mathrm{i} y)=\sin x \cosh y+\mathrm{i} \cos x \sinh y \end{aligned} $$ In the special case \(x=0, y \in \mathbb{R}\) we have $$ \cos (\mathrm{i} y)=\frac{1}{2}\left(e^{y}+e^{-y}\right)=\cosh y \quad \text { and } \quad \sin (\mathrm{i} y)=\frac{\mathrm{i}}{2}\left(e^{y}-e^{-y}\right)=\mathrm{i} \sinh y $$ Determine all the \(z \in \mathbb{C}\) with \(|\sin z| \leq 1\), and find an \(n \in \mathbb{N}\) such that $$ |\sin (\mathrm{in})|>10000. $$

Step-by-Step Solution

Answer

All complex numbers \(z\) such that \(|\sin z| \leq 1\) are identified by \(\sin^2 x + \sinh^2 y \leq 1\). For \(n = 10\), \(|\sin(\mathrm{i}n)| > 10000\).

Step 1: Understand the Task

We need to analyze the complex functions and determine all complex numbers \(z\) such that \(|\sin z| \leq 1\). Additionally, find a natural number \(n\) such that \(|\sin(\mathrm{i}n)| > 10000\).

Step 2: Use Given Formulas for \(\cos z\) and \(\sin z\)

From the given, we have:\[sin z = \sin(x + \mathrm{i} y) = \sin x \cosh y + \mathrm{i} \cos x \sinh y\]

Step 3: Analyze \(|\sin z|\)

The modulus of \(\sin z\) is:\[|\sin z| = |\sin x \cosh y + \mathrm{i} \cos x \sinh y| = \sqrt{(\sin x \cosh y)^2 + (\cos x \sinh y)^2}\]

Step 4: Simplify and Set the Condition \(|\sin z| \leq 1\)

We have:\[(\sin x \cosh y)^2 + (\cos x \sinh y)^2 \leq 1\]which simplifies to\[\sin^2 x \cosh^2 y + \cos^2 x \sinh^2 y \leq 1\]

Step 5: Apply Trigonometric Identity

Using the identity \(\sin^2 x + \cos^2 x = 1\), we substitute:\[\cosh^2 y = 1 + \sinh^2 y\]Thus,\[\sin^2 x(1 + \sinh^2 y) + \cos^2 x \sinh^2 y \leq 1\]This simplifies to\[\sin^2 x + \sinh^2 y \leq 1\]

Step 6: Consider the Case \(z = \mathrm{i}y\)

Now consider \(z = \mathrm{i}y\), \(x = 0\). Then \(\sin(\mathrm{i}y) = \mathrm{i}\sinh y\) and \(\cos(\mathrm{i}y) = \cosh y\). The modulus becomes:\[|\sin(\mathrm{i}y)| = |\mathrm{i} \sinh y| = |\sinh y|\] We need to find when \(|\sinh y| > 10000\).

Step 7: Solve \(|\sinh y| > 10000\)

\(\sinh y\) is defined as:\[\sinh y = \frac{e^y - e^{-y}}{2}\]We want:\[|\sinh y| = \frac{|e^y - e^{-y}|}{2} > 10000\]

Step 8: Find \(n\) Where \(|\sinh n| > 10000\)

We find \(n\) such that:\[\left|\frac{e^y - e^{-y}}{2}\right| > 10000\]For larger \(n\), \(e^n \approx e^y\) and \(\sinh y \approx \frac{e^y}{2} > 10000\).Hence, \(e^y > 20000\). Taking \(y = n\), we solve \(e^n > 20000\), thus approximate \(n \approx \ln 20000 \approx 9.9035\). Rounding up gives \(n = 10\).

Key Concepts

Complex Numbers

Complex numbers are an extension of the real numbers, formed by combining a real number with an imaginary number. A complex number is typically written in the form of \(z = x + \mathrm{i}y\), where \(x\) and \(y\) are real numbers, and \(\mathrm{i}\) is the imaginary unit with the property \(\mathrm{i}^2 = -1\). Complex numbers are incredibly useful in various fields, such as electrical engineering, fluid dynamics, and quantum physics.
The real part of the complex number \(z\) is \(x\), and the imaginary part is \(y\). One of the fascinating properties of complex numbers is the ability to conjugate them. The conjugate of \(z\) is written as \(\bar{z} = x - \mathrm{i}y\). This operation flips the sign of the imaginary part, and it is handy when dealing with complex exponentials and trigonometric functions since it helps in simplifying the expressions and maintaining the equality, as seen in the exercise formulas for exponential, sine, and cosine functions for complex numbers.

Trigonometric Functions

Trigonometric functions like sine and cosine are vital in the realm of complex analysis. When extended to complex numbers, they reveal rich and complex behavior based on Euler's formula. For a complex number \(z = x + \mathrm{i}y\), the sine function can be expressed as \(\sin(x + \mathrm{i}y) = \sin x \cosh y + \mathrm{i} \cos x \sinh y\).
Similarly, the cosine function takes the form \(\cos(x + \mathrm{i}y) = \cos x \cosh y - \mathrm{i} \sin x \sinh y\).
These expressions highlight the mix of real trigonometric functions with hyperbolic ones, extending their use far beyond triangles and circles.
This mixing leads to practical applications in wave equations and signal processing, illustrating how complex trigonometric functions are vastly more general than their real counterparts.

Hyperbolic Functions

Hyperbolic functions, often compared to trigonometric counterparts, play a crucial role when extending real functions into the complex plane. Functions like \(\sinh\) and \(\cosh\), despite their resemblance to sine and cosine, relate to hyperbolas rather than circles.
For complex numbers, these functions can be described using exponentials:\(\cosh y = \frac{e^y + e^{-y}}{2}\) and \(\sinh y = \frac{e^y - e^{-y}}{2}\).
These definitions reveal how hyperbolic functions arise naturally when dealing with complex exponential functions, providing seamless interpretations even into higher dimensions.
This is particularly useful as seen in the calculations for complex numbers with the imaginary unit, where hyperbolic functions intertwine with trigonometric functions, showing their complementary nature.

Inequalities

Inequalities play a pivotal role in mathematics as they establish relative order and offer bound conditions that are essential for problem-solving. In complex analysis, dealing with inequalities often means working with the modulus of complex numbers.
In the given exercise, the inequality \(|\sin z| \leq 1\) poses an interesting challenge that involves understanding the boundaries within which complex numbers must lie for the condition to hold. By analyzing the trigonometric expressions of complex sine:\(\sin(x + \mathrm{i}y) = \sin x \cosh y + \mathrm{i} \cos x \sinh y\),we conclude that \(\sin^2 x + \sinh^2 y \leq 1\).This ensures the modulus, a measure similar to distance, remains within predefined limits, helping identify valid complex numbers.

Modulus of Complex Numbers

The modulus of a complex number \(z = x + \mathrm{i}y\) is a measure of its distance from the origin in the complex plane and is denoted by \(|z|\). It is calculated as \(|z| = \sqrt{x^2 + y^2}\).
For problems involving complex functions, like sine, the modulus gives a way to understand its magnitude or absolute value. For the function \(\sin z\), since \(|\sin z| = \sqrt{(\sin x \cosh y)^2 + (\cos x \sinh y)^2}\),we notice how both trigonometric and hyperbolic functions merge.
This concept is crucial as it enables us to tackle inequalities, such as ensuring that this modulus stays within certain bounds, making it possible to determine plausible solutions to complex equations.