Problem 25 - Advanced Engineering Mathematics

Question

$$ \text { Prove that } \cosh z=\cosh x \cos y+i \sinh x \sin y \text {. } $$

Step-by-Step Solution

Answer

The identity \( \cosh z = \cosh x \cos y + i \sinh x \sin y \) is proven using the definitions of hyperbolic and trigonometric identities.

Step 1: Recall the definition of hyperbolic cosine

The hyperbolic cosine function is defined as \( \cosh z = \frac{e^z + e^{-z}}{2} \). This identity is central to proving the given equation.

Step 2: Express \( z \) in terms of real and imaginary parts

In the complex number \( z = x + iy \), \( x \) and \( y \) are real numbers. Express \( e^z \) as \( e^{x+iy} = e^x e^{iy} \).

Step 3: Simplify \( e^z \) using Euler's formula

Using Euler's formula, \( e^{iy} = \cos y + i \sin y \). Therefore, \( e^{x+iy} = e^x (\cos y + i \sin y) \). Similarly, \( e^{-z} = e^{-x} (\cos y - i \sin y) \) as derived from \( e^{-z} = e^{-(x+iy)} = e^{-x}e^{-iy} \).

Step 4: Compute \( \cosh z \) with derived components

Substitute back into the definition: \[ \cosh z = \frac{e^x (\cos y + i \sin y) + e^{-x} (\cos y - i \sin y)}{2} \].

Step 5: Distribute and Combine Terms

Distribute and combine real and imaginary terms: \[ \Rightarrow \cosh z = \frac{e^x \cos y + i e^x \sin y + e^{-x} \cos y - i e^{-x} \sin y}{2} \]. Combine like terms: \[ = \frac{(e^x + e^{-x}) \cos y}{2} + i \frac{(e^x - e^{-x}) \sin y}{2} \].

Step 6: Simplify using identities for hyperbolic and trigonometric functions

Recall that \( \cosh x = \frac{e^x + e^{-x}}{2} \) and \( \sinh x = \frac{e^x - e^{-x}}{2} \). Substitute these identities back to get: \[ \cosh z = \cosh x \cos y + i \sinh x \sin y \].

Step 7: Conclusion

We obtained that \( \cosh z \) simplifies exactly to \( \cosh x \cos y + i \sinh x \sin y \), proving the given equation.

Key Concepts

Hyperbolic Functions

Hyperbolic functions are analogs of trigonometric functions but for a hyperbola rather than a circle. They have important applications in areas such as calculus, complex analysis, and many engineering fields. The two most basic hyperbolic functions are hyperbolic sine (\( \,\sinh z\, \)) and hyperbolic cosine (\( \,\cosh z\, \)).

• Hyperbolic cosine, \(\cosh z\), is defined as \(\cosh z = \frac{e^z + e^{-z}}{2}\).
• Hyperbolic sine, \(\sinh z\), is defined as \(\sinh z = \frac{e^z - e^{-z}}{2}\).

The addition and subtraction of the exponential terms, often related to Euler’s formula, underpin these functions' behavior.
They are crucial when dealing with hyperbolas, much like how sine and cosine relate to circles. One key relationship to note is the identity:\[ \cosh^2 z - \sinh^2 z = 1 \]which mimics the trigonometric Pythagorean identity \(\cos^2 x + \sin^2 x = 1\).

Euler's Formula

Euler's formula is a fundamental bridge between the exponential function and trigonometric functions. It is typically expressed as \( e^{ix} = \cos x + i \sin x \).
This relationship elegantly shows how complex numbers interact with rotations in the plane. It's deeply linked to the complex exponential function and facilitates expressing periodic wave functions in physics and engineering.
In the given problem, Euler's formula allows us to break down complex exponents:

• For complex number \(z = x + iy\), we use \(e^{iy} = \cos y + i \sin y\).
• Consequently, any expression involving \(e^{z}\) or \(e^{-z}\) can be decomposed using Euler's insights.

Using this, complex expressions can be rewritten in terms of more familiar trigonometric forms,
thereby simplifying calculations and understanding of phenomena such as oscillations and waves.

Complex Numbers

Complex numbers extend the real numbers with a unique component: \(i\), the imaginary unit, satisfying \(i^2 = -1\). Each complex number has the form \(z = x + iy\), where \(x\) is the real part and \(y\) is the imaginary part.

The capability to write real and imaginary components allows complex numbers to be manipulated like vectors. Operations such as addition, multiplication, and exponentiation can be performed under rules extending those for real numbers.

• Adding or subtracting involves managing real and imaginary parts separately — \((x_1 + iy_1) + (x_2 + iy_2) = (x_1 + x_2) + i(y_1 + y_2)\).
• Multiplication is done through distributive law: \((x + iy)(u + iv) = (xu - yv) + i(xv + yu)\).

In complex analysis and hyperbolic functions, complex numbers allow definitions and manipulations to extend naturally onto the complex plane. They provide substantial insight into oscillations, electrical engineering, and signal processing, which rely heavily on these foundational properties.