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) \(\cos z=\cos (x+i y)=\cos x \cosh y-i \sin x \sinh y\) \(\sin z=\sin (x+i y)=\sin x \cosh y+i \cos x \sinh y\) 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 $$ |\sin (\mathrm{i} n)|>10000 $$

Step-by-Step Solution

Answer

For \(|\sin z| \leq 1\), set \(x=0\) and \(y\) such that \(|\sinh y| \leq 1\). Choose \(n=10\) for \(|\sin(in)| > 10000\).

Step 1: Interpret the Problem

We need to find complex numbers \(z = x + iy\) such that \(|\sin z| \leq 1\) and then find an integer \(n\) such that \(|\sin (in)| > 10000\). From part (b), we have formulas for \(\sin z\) and \(\cos z\).

Step 2: Analyze \(\sin z\) Expression

From the formula \(\sin z = \sin(x + iy) = \sin x \cosh y + i \cos x \sinh y\), calculate the magnitude: \[ | ext{Im}(\sin z)| = |\cos x \sinh y| \]and \[ |\sin z| = \sqrt{(\sin x \cosh y)^2 + (\cos x \sinh y)^2} \] We need \(|\sin z| \leq 1\).

Step 3: Set \(x = 0\) for Simplification

For simplicity, set \(x = 0\) so that \(\sin x = 0\) and \(\cos x = 1\). Then, \[ \sin z = i \sinh y \] and \[ |\sin z| = |i \sinh y| = |\sinh y| \leq 1 \].

Step 4: Solve \(|\sinh y| \leq 1\)

Recall that \(\sinh y = \frac{e^y - e^{-y}}{2}\). Determine \(y\) so that:\[ |\sinh y| = \left|\frac{e^y - e^{-y}}{2}\right| \leq 1 \].This means solving for real \(y\) such that \(-1 \leq \sinh y \leq 1\).

Step 5: Analyze \(\sin(iy)\) for Large Values

Using the result from special case when \(x=0\), we have \[ \sin(iy) = i \sinh y \].To find \(n\) such that \(|\sin(in)| > 10000\) substitute \(y = n\) and solve:\(|\sinh n| > 10000\).

Step 6: Solve \(|\sinh n| > 10000\)

We need \(\sinh n > 10000\), hence,\[ \frac{e^n - e^{-n}}{2} > 10000 \]. Approximately, for large \(n\), this becomes \(e^n > 20000\). Solve for \(n\) by:\[ n > \ln(20000) \].

Step 7: Calculate \(n\)

By estimating, we find:\[ n \approx \ln(20000) \approx \ln(10^4.30103) = 4.30103 \times \ln(10) = 4.30103 \times 2.302 = 9.904 \].Therefore, take \(n=10\) since \(n\) must be an integer.

Key Concepts

Complex Exponential Functions

Complex exponential functions extend the concept of exponential functions into the complex plane. When dealing with complex numbers of the form \(z = x + iy\), the exponential function is represented as \(\exp(z) = e^{x+iy}\). This can be broken down using Euler's formula, \(e^{ix} = \cos x + i\sin x\). Consequently, we express the complex exponential function as:

• \( e^{iy} = \cos y + i\sin y \)
• \( e^{z} = e^{x}(\cos y + i\sin y) \)

This is a powerful way to connect complex numbers and trigonometric functions.
Moreover, the exponential function has properties that include the conjugate identity \(\overline{\exp(z)} = \exp(\overline{z})\), which shows the symmetry and consistency in extending real functions to the complex domain.

Hyperbolic Functions

Hyperbolic functions can be seen as analogs of trigonometric functions but for the hyperbola, rather than the circle. They are defined as combinations of exponential functions. The two main hyperbolic functions are the hyperbolic sine and cosine:

• \( \sinh y = \frac{e^y - e^{-y}}{2} \)
• \( \cosh y = \frac{e^y + e^{-y}}{2} \)

Hyperbolic functions appear frequently in equations involving complex numbers. For example, the formulas \(\cos z = \cos x \cosh y - i \sin x \sinh y\) and \(\sin z = \sin x \cosh y + i \cos x \sinh y\) illustrate how hyperbolic functions are intertwined with complex arguments like \(z = x + iy\).
These functions prove crucial in understanding the behavior of complex numbers, especially when analyzing their trigonometric identities.

Magnitude of Complex Numbers

The magnitude (or modulus) of a complex number \(z = x + iy\) is the distance of the point \(z\) from the origin in the complex plane. It is given by the formula \(|z| = \sqrt{x^2 + y^2}\). Similar principles apply to the magnitude of complex functions such as \(\sin(z)\), where \(|\sin z|\) is computed using the real and imaginary parts obtained through trigonometric and hyperbolic identities.
For \(\sin z = \sin x \cosh y + i \cos x \sinh y\), the magnitude can be calculated as:

• \(|\sin z| = \sqrt{(\sin x \cosh y)^2 + (\cos x \sinh y)^2}\)

By using these calculations, we can undertake problems that involve constraints like \(|\sin z| \leq 1\), as discussed in the exercise. This notion of magnitude is fundamental in solving complex equations and analyzing complex functions.