Problem 31 - Advanced Engineering Mathematics

Question

$$ \text { Prove that } \tanh z \text { is periodic with period } \pi i \text {. } $$

Step-by-Step Solution

Answer

\( \tanh z \) is periodic with period \( \pi i \).

Step 1: Understand the Definition of Hyperbolic Tangent

The hyperbolic tangent function is defined as \( \tanh z = \frac{\sinh z}{\cosh z} \), where \( \sinh z = \frac{e^z - e^{-z}}{2} \) and \( \cosh z = \frac{e^z + e^{-z}}{2} \). Understanding these definitions is crucial for determining periodicity.

Step 2: Calculate \( \tanh(z + \pi i) \)

We need to show that \( \tanh z = \tanh(z + \pi i) \). Calculate \( \sinh(z + \pi i) = \frac{e^{z+\pi i} - e^{-z-\pi i}}{2} \) and \( \cosh(z + \pi i) = \frac{e^{z+\pi i} + e^{-z-\pi i}}{2} \). Simplify these using Euler's formula: \( e^{z+\pi i} = -e^z \) and \( e^{-z-\pi i} = -e^{-z} \).

Step 3: Simplify \( \sinh(z + \pi i) \) and \( \cosh(z + \pi i) \)

Using the identities from Step 2, simplify:\[ \sinh(z + \pi i) = \frac{-e^z - (-e^{-z})}{2} = \frac{-e^z + e^{-z}}{2} = -\sinh z \]\[ \cosh(z + \pi i) = \frac{-e^z + (-e^{-z})}{2} = \frac{-e^z - e^{-z}}{2} = -\cosh z \]

Step 4: Compare \( \tanh z \) and \( \tanh(z + \pi i) \)

Using the simplified expressions from Step 3, find \( \tanh(z + \pi i) = \frac{-\sinh z}{-\cosh z} = \frac{\sinh z}{\cosh z} = \tanh z \). This confirms that \( \tanh(z + \pi i) = \tanh z \).

Step 5: Conclude the Periodicity

Since \( \tanh(z + \pi i) = \tanh z \), we have shown that \( \tanh z \) is periodic with period \( \pi i \). This means that the function repeats its values along the imaginary axis every \( \pi i \) units.

Key Concepts

Hyperbolic Functions

Hyperbolic functions, although similar in name to trigonometric functions, differ as they are based on exponential functions. These functions include \(\sinh\) and \(\cosh\), which contribute significantly to the study of complex analysis. \(
\)\(\sinh z\) and \(\cosh z\) are defined as follows:

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

\(
\)The hyperbolic tangent \(\tanh z\) is expressed as the ratio of \(\sinh z\) to \(\cosh z\), resulting in the formula \(\tanh z = \frac{\sinh z}{\cosh z}\). These functions are essential in demonstrating the periodic nature of complex variables, and they illustrate fascinating behaviors in the complex plane.

Complex Periodicity

In the realm of complex analysis, understanding periodicity is crucial. Periodicity refers to the property of a function repeating its values at regular intervals. For the function \(\tanh z\), the periodicity becomes evident when explored thoroughly. It means that moving a certain distance along the imaginary axis does not change the function's value. This is particularly intriguing in complex analysis because it involves the imaginary unit \(i\).\(
\)For \(\tanh z\), we establish its periodicity by examining \(\tanh(z + \pi i)\). By calculating and simplifying to find \(\tanh(z + \pi i) = \tanh z\), we ascertain that \(\tanh z\) repeats itself every \(\pi i\) along the imaginary axis. This result illustrates the beautiful and robust nature of complex functions and their periodic properties.

Euler's Formula

Euler's formula is a significant cornerstone in mathematics, particularly within complex analysis. It provides a bridge between exponential functions and trigonometric functions. Euler's formula states that for any real number \(\theta\), \(e^{i\theta} = \cos \theta + i \sin \theta\). This formula extends beautifully into the complex plane, allowing for various complex identities and simplifying complex expressions.\(
\)In the context of the problem, Euler's formula assists in simplifying expressions involving complex exponents. When showing \(\tanh z\) as periodic, Euler's formula reveals that \(e^{z+\pi i} = -e^z\) and \(e^{-z-\pi i} = -e^{-z}\). This simplification is crucial in confirming that the function's value remains unchanged when shifted by \(\pi i\) along the imaginary axis.\(
\)Euler's formula is more than just a mathematical identity. It offers profound insights into the behavior of complex exponential functions, linking seemingly disparate areas of mathematics into a cohesive whole.