Problem 31 - Advanced Engineering Mathematics

Question

$$\begin{aligned} \tanh (z+\pi i) &=\frac{\sinh (x+(y+\pi) i)}{\cosh (x+(y+\pi) i)}=\frac{\sinh x \cos (y+\pi)+i \cosh x \sin (y+\pi)}{\cosh x \cos (y+\pi)+i \sinh x \sin (y+\pi)} \\ &=\frac{-[\sinh x \cos y+i \cosh x \sinh y]}{-[\cosh x \cos y+i \sinh x \sin y]}=\frac{-\sinh z}{-\cosh z}=\tanh z \end{aligned}$$

Step-by-Step Solution

Answer

The expression \( \tanh(z + \pi i) = \tanh(z) \).

Step 1: Understand the Expression

We are given the function \( \tanh(z + \pi i) \) and its parts involving hyperbolic sine `\( \sinh \)` and hyperbolic cosine `\( \cosh \)`. Our goal is to show that it is equal to \( \tanh(z) \).

Step 2: Analytically Express \(\sinh\) and \(\cosh\)

The hyperbolic functions can be represented in terms of exponential functions as follows: \( \sinh(x + yi) = \sinh x \cos(y + \pi) + i \cosh x \sin(y + \pi) \) and \( \cosh(x + yi) = \cosh x \cos(y + \pi) + i \sinh x \sin(y + \pi) \).

Step 3: Simplify Cosine and Sine Components

Using the identities for trigonometric functions, we know \( \cos(y + \pi) = -\cos(y) \) and \( \sin(y + \pi) = -\sin(y) \). Substitute these into the expressions for the components of \( \sinh \) and \( \cosh \).

Step 4: Substitute and Simplify

By substituting and simplifying, we obtain \( \sinh(x + (y + \pi)i) = -\sinh x \cos y - i \cosh x \sin y \) and \( \cosh(x + (y + \pi)i) = -\cosh x \cos y - i \sinh x \sin y \).

Step 5: Evaluate \( \frac{-\sinh z}{-\cosh z} \)

The expression simplifies to \( \frac{-\sinh z}{-\cosh z} \), which is \( \tanh z \) because the negatives cancel each other.

Key Concepts

Hyperbolic Functions

Hyperbolic functions, like \(\sinh\) and \(\cosh\), are analogous to trigonometric functions but are defined using exponential functions. They have unique properties and use cases, especially in complex analysis:

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

These functions arise naturally when dealing with hyperbolic geometry and appear in solutions to hyperbolic differential equations.
When extending these functions to complex numbers, we handle them in terms of both real and imaginary components. This is especially important in complex analysis, which often requires the use of hyperbolic functions within the domain of complex numbers. This comes into play during transformations and while evaluating complex expressions.

Analytic Expressions

Analytic expressions involve finding a way to rewrite mathematical functions in terms of more fundamental operations. For hyperbolic functions, this often implies re-casting them in the form of exponential functions.
In our exercise, we are given the function \(\tanh(z + \pi i)\), which uses the identities:

• \(\sinh(x + yi) = \sinh x \cos(y + \pi) + i \cosh x \sin(y + \pi)\)
• \(\cosh(x + yi) = \cosh x \cos(y + \pi) + i \sinh x \sin(y + \pi)\)

Here, the variables \(x\) and \(y\) represent real numbers and \(i\) is the imaginary unit.
The analytic expression for hyperbolic functions in complex form helps articulate their behavior over different values of \(z\), a complex variable.

Trigonometric Identities

Trigonometric identities are crucial tools that simplify expressions involving trigonometric functions. In this context, these identities help transition expressions into solvable forms by identifying equivalent expressions.
For instance, identities like:

• \(\cos(y + \pi) = -\cos y\)
• \(\sin(y + \pi) = -\sin y\)

were used in the problem to transform the components of \(\sinh\) and \(\cosh\) functions. These transformations are vital because they allow other equations to be plugged into or simplified, revealing the simplicity underlying complex structures.
In essence, being able to toggle between different trigonomic forms enables the derivation of clean solutions.

Simplification Techniques

Simplification techniques are methods that reduce complex expressions into more manageable forms. Applying these techniques requires the use of mathematical identities and properties.
In the original exercise, simplification occurs through:

• Substituting trigonometric identities into hyperbolic forms.
• Recognizing that both the numerator and denominator of \(\frac{\sinh z}{\cosh z}\) are negated, allowing those negatives to cancel.

The result of these simplifications is an elegant expression, \(\tanh z\), showing that despite initial complexity, these operations have interconnections that simplify the problem.
These techniques highlight an important aspect of mathematical analysis: finding pathways through which complicated problems can be navigated towards simpler forms.