Problem 24 - Advanced Engineering Mathematics

Question

Prove that \(\sinh z=\sinh x \cos y+i \cosh x \sin y\)

Step-by-Step Solution

Answer

The proof shows \( \sinh z = \sinh x \cos y + i \cosh x \sin y \) using hyperbolic and exponential identities.

Step 1: Understand Hyperbolic Sine Definition

Recall that the hyperbolic sine function is defined as \( \sinh z = \frac{e^z - e^{-z}}{2} \). Here, \( z \) is a complex number expressed as \( z = x + iy \), where \( x \) and \( y \) are real numbers.

Step 2: Express Complex Exponentials

Substitute \( z = x + iy \) into the hyperbolic sine definition. We have \( \sinh(x + iy) = \frac{e^{x+iy} - e^{-(x+iy)}}{2} \).

Step 3: Apply Euler's Formula

Utilize Euler's formula, \( e^{ix} = \cos x + i \sin x \), to express the complex exponentials: \( e^{x+iy} = e^x (\cos y + i \sin y) \) and \( e^{-(x+iy)} = e^{-x} (\cos(-y) + i \sin(-y)) = e^{-x} (\cos y - i \sin y) \).

Step 4: Substitute and Simplify

Substitute the expressions from Step 3 into the equation for \( \sinh(x + iy) \). This gives \( \sinh(x + iy) = \frac{e^x(\cos y + i \sin y) - e^{-x}(\cos y - i \sin y)}{2} \).

Step 5: Collect Real and Imaginary Terms

Separate the terms within the numerator: \( e^x \cos y - e^{-x} \cos y + i(e^x \sin y + e^{-x} \sin y) \). Factor out common terms: \( (e^x - e^{-x}) \cos y + i(e^x + e^{-x}) \sin y \).

Step 6: Use Hyperbolic Identities

Recognize and apply the hyperbolic identities \( e^x - e^{-x} = 2 \sinh x \) and \( e^x + e^{-x} = 2 \cosh x \). Substitute into the expression to get \( 2 \sinh x \cos y + i(2 \cosh x \sin y) \).

Step 7: Finalize Expression

Divide the obtained expression by 2: \( \sinh x \cos y + i \cosh x \sin y \). This is the required proof that \( \sinh z = \sinh x \cos y + i \cosh x \sin y \).

Key Concepts

Hyperbolic Functions

Hyperbolic functions offer an intriguing parallel to the familiar trigonometric functions. Frequently encountered in different branches of mathematics, hyperbolic functions, such as hyperbolic sine (\( \sinh z \)) and hyperbolic cosine (\( \cosh z \)), are crucial for complex analysis. These functions are defined using exponential functions, capturing their unique nature:

• The hyperbolic sine function is given by:\[ \sinh z = \frac{e^z - e^{-z}}{2} \]
• The hyperbolic cosine function is defined as:\[ \cosh z = \frac{e^z + e^{-z}}{2} \]

Both functions have significant roles in mathematics, as they appear in solutions to differential equations and are associated with the geometry of hyperbolic spaces.
In general, they work well with complex numbers and are closely tied to Euler's formula, paving the way to understand how complex exponentials play into hyperbolic identities.

Euler's Formula

Euler's formula is an essential bridge in complex analysis. It links complex exponentials with trigonometric functions. Euler's formula states that for any real number \( x \):

• \[ e^{ix} = \cos x + i \sin x \]

It's a profound tool showing how complex numbers relate to cycles, waves, and rotations. In solving problems like the given exercise, Euler's formula helps express exponentials of complex numbers in a much more digestible format. By utilizing Euler's formula, we can simplify the terms \( e^{x+iy} \) and \( e^{-(x+iy)} \), making complex analysis more approachable and revealing the trigonometric nature hidden within complex expressions. This formula showcases the seamless integration between exponential functions and rotations, offering invaluable insight into the structure of complex numbers.

Complex Numbers

Complex numbers are a fundamental component in many areas of mathematics, including complex analysis. A complex number \( z \) generally takes the form of \( z = x + iy \), where \( x \) represents the real part and \( y \) stands for the imaginary part. Here,

• The unit imaginary number \( i \) satisfies \( i^2 = -1 \).

Understanding complex numbers involves not only grasping their algebraic structure but also recognizing their geometric representation. They can be visualized as points or vectors in a two-dimensional plane, with the horizontal axis representing the real component and the vertical axis portraying the imaginary part.
This visualization aids in interpreting operations like addition, subtraction, and multiplication of complex numbers.
In the context of hyperbolic functions, complex numbers allow the exploration of their behavior beyond real arguments, unlocking elegant connections such as those demonstrated in the exercise problem.

Hyperbolic Identities

Hyperbolic identities are akin to trigonometric identities but pertain to hyperbolic functions. These identities provide relationships that are essential for simplifying expressions involving hyperbolic functions. Some fundamental hyperbolic identities include:

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

These identities mirror the properties and relationships found in trigonometry but apply to the hyperbolic functions' realm. They are pivotal in proving and simplifying expressions that involve hyperbolic functions, especially when dealing with complex arguments. In the exercise, these identities were crucial to separate real and imaginary components effectively, leading to the final elegant expression:\[ \sinh z = \sinh x \cos y + i \cosh x \sin y \]Utilizing such identities simplifies complex expressions and helps to unveil deeper symmetrical structures within mathematical expressions, contributing to a more profound understanding of these functions.