Problem 27 - Advanced Engineering Mathematics

Question

Prove that \(|\cosh z|^{2}=\cos ^{2} y+\sinh ^{2} x\).

Step-by-Step Solution

Answer

Using complex analysis and hyperbolic identities, it holds that \(|\cosh z|^2 = \cos^2 y + \sinh^2 x\).

Step 1: Recall the Definition of Hyperbolic Cosine

The hyperbolic cosine function is defined as \( \cosh z = \frac{e^z + e^{-z}}{2} \) for any complex number \( z \). If \( z = x + yi \), we need to express \( \cosh z \) using this definition.

Step 2: Substitute for a Complex Number

Let \( z = x + yi \), where \( x, y \) are real numbers. Then, \[ \cosh z = \frac{e^{x+yi} + e^{-(x+yi)}}{2} = \frac{e^x e^{yi} + e^{-x} e^{-yi}}{2}. \]

Step 3: Use Euler's Formula

By Euler's formula, \( e^{yi} = \cos y + i \sin y \) and \( e^{-yi} = \cos y - i \sin y \). Substitute these into \( \cosh z \): \[ \cosh z = \frac{e^x (\cos y + i \sin y) + e^{-x} (\cos y - i \sin y)}{2}. \]

Step 4: Simplify \( \cosh z \) Using Trigonometric Identities

Simplify the expression by combining real and imaginary parts:\[\cosh z = \frac{(e^x + e^{-x}) \cos y + i(e^x - e^{-x}) \sin y}{2}.\]This gives us:\[ \cosh z = \cosh x \cos y + i \sinh x \sin y. \]

Step 5: Find \( |\cosh z|^2 \)

The magnitude squared of a complex number \( a + bi \) is \( a^2 + b^2 \). Apply this to \( \cosh z = \cosh x \cos y + i \sinh x \sin y \):\[|\cosh z|^2 = (\cosh x \cos y)^2 + (\sinh x \sin y)^2.\]

Step 6: Expand the Expression

Simplify \((\cosh x \cos y)^2 + (\sinh x \sin y)^2\) to \[|\cosh z|^2 = \cosh^2 x \cos^2 y + \sinh^2 x \sin^2 y.\]

Step 7: Use Identity \(\cosh^2 x - \sinh^2 x = 1\)

Using the identity \( \cosh^2 x - \sinh^2 x = 1 \), note that \( \cosh^2 x = 1 + \sinh^2 x \). Thus, substitute it back:\[\cosh^2 x \cos^2 y = (1 + \sinh^2 x) \cos^2 y = \cos^2 y + \sinh^2 x \cos^2 y.\]

Step 8: Combine and Arrive at Final Expression

Thus, whether or not the identity statement holds, we re-arrive after simplification to confirm:\[|\cosh z|^2 = \cos^2 y + \sinh^2 x \cos^2 y + \sinh^2 x \sin^2 y.\]By factoring, based on the well-known identity: \(sin^2 y + \cos^2 y = 1\), then it simplifies and we regain that \[|\cosh z|^2 = \cos^2 y + \sinh^2 x.\]

Key Concepts

Hyperbolic Functions

Hyperbolic functions are analogs of the trigonometric functions but for a hyperbola, much like how the usual trigonometric functions relate to a circle. The two primary hyperbolic functions are the hyperbolic sine, \(\sinh(x)\), and the hyperbolic cosine, \(\cosh(x)\).
They are defined using exponential functions as follows:

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

The hyperbolic cosine function, \(\cosh(x)\), shares a key identity with the hyperbolic sine, which is \(\cosh^2(x) - \sinh^2(x) = 1\). This identity closely resembles the Pythagorean identity found in trigonometry, \(\cos^2(x) + \sin^2(x) = 1\). Understanding these functions is crucial for exploring complex exponential expressions and identities.

Euler's Formula

Euler's formula provides a deep connection between complex numbers and trigonometry. The formula states that for any real number \(y\), \(e^{yi} = \cos(y) + i\sin(y)\).
This demonstrates that complex exponentials can be expressed in terms of sinusoidal functions. Similarly, \(e^{-yi} = \cos(y) - i\sin(y)\), which complements the original formula.

Application in Complex Analysis

Euler's formula is pivotal when dealing with expressions involving complex exponentials, just like in the problem we're analyzing. It simplifies expressions by converting them into sinusoids, aiding in easier manipulation and understanding of the equation.
By applying Euler's formula, any exponential function of a complex number \(x + yi\) can be easily broken down into real and imaginary components. This is integral for dealing with problems involving hyperbolic functions of complex numbers, specifically those involving mixed real and imaginary inputs.

Trigonometric Identities

Trigonometric identities are mathematical equations that relate the angles and ratios of trigonometric functions, serving as tools for simplifying complex expressions. Key identities include:

• Pythagorean Identity: \(\cos^2(y) + \sin^2(y) = 1\)
• Angle Sum and Difference Identities
• Double Angle and Half Angle Formulas

In complex analysis, these identities help in manipulating and simplifying expressions involving trigonometric and hyperbolic functions. For instance, the Pythagorean identity \(\cos^2(y) + \sin^2(y) = 1\) is particularly useful when working with expressions formatted via Euler's formula.

Role in Complex Analysis

Using trigonometric identities allows for precision in simplifying mixed exponential and trigonometric problems. As seen in the solution, recognizing and utilizing these identities lets us transition from complex exponential expressions back to simpler algebraic terms, ultimately aiding in confirming complex identities.

Complex Numbers

Complex numbers are fundamental in complex analysis, comprising a real part and an imaginary part, expressed as \(z = x + yi\). Here, \(x\) represents the real component, while \(y\) represents the imaginary component, with \(i\) being the imaginary unit satisfying \(i^2 = -1\).
Complex numbers are essential as they provide a broader number system where certain algebraic problems have solutions that do not exist within the real numbers alone.

Magnitude of Complex Numbers

The magnitude, or modulus, of a complex number \(z = a + bi\) is given by \(\|z\| = \sqrt{a^2 + b^2}\). This magnitude represents the distance of the complex number from the origin on the complex plane.

Application in Hyperbolic Functions

In our problem context, determining the magnitude involves the complex exponential forms of hyperbolic functions like \(\cosh(z)\). Finding the magnitude of \(\cosh(z)\) especially involves combining the real and imaginary parts to ascertain \(|\cosh(z)|^2 = a^2 + b^2\). Recognizing interactions among complex numbers, and utilizing their properties, allows for proving expressions and validating identities like \(|\cosh z|^2 = \cos^2 y + \sinh^2 x\).