Problem 17 - Advanced Engineering Mathematics

Question

Find all values of \(z\) satisfying the given equation. \(\sinh z=-i\)

Step-by-Step Solution

Answer

The solutions are \(z = \ln(\sqrt{2} - 1) + i\left(\frac{\pi}{2} + n\pi\right)\) for integer \(n\).

Step 1: Understand the properties of hyperbolic sine

The hyperbolic sine function, \(\sinh z\), is defined as \(\sinh z = \frac{e^z - e^{-z}}{2}\). It's important to convert the given equation into terms that match this formula.

Step 2: Set up the equation in terms of exponential form

We need to solve \(\frac{e^z - e^{-z}}{2} = -i \). Clear the fraction by multiplying both sides by 2, obtaining \( e^z - e^{-z} = -2i \).

Step 3: Utilize the identity for \(e^{z}\)

Express \(z\) as \(x + iy\), where \(x\) is the real part and \(y\) is the imaginary part. Then \(e^z = e^x e^{iy} = e^x(\cos(y) + i\sin(y))\).

Step 4: Substitute into the equation

Substituting \(e^z\) and \(e^{-z}\) into \(e^z - e^{-z} = -2i\), we get: \[ e^x(\cos(y) + i\sin(y)) - e^{-x}(\cos(y) - i\sin(y)) = -2i \].

Step 5: Separate real and imaginary parts

Separate the equation into real and imaginary parts, resulting in two equations: \[ e^x \cos(y) - e^{-x} \cos(y) = 0 \] and \[ e^x \sin(y) + e^{-x} \sin(y) = -2 \].

Step 6: Simplify the real part

From the real part, factor out \(\cos(y)\) yielding \(\cos(y)(e^x - e^{-x}) = 0\). This implies either \(\cos(y) = 0\) or \(e^x - e^{-x} = 0\).

Step 7: Solve based on \(\cos(y) = 0\)

If \(\cos(y) = 0\), then \(y = (2n+1)\frac{\pi}{2}\), where \(n\) is an integer. Substitute into the imaginary equation to find \(x\).

Step 8: Analyze specific solutions for \(y\)

Suppose \(y = \frac{\pi}{2}\), then \(e^x \sin(y) + e^{-x} \sin(y) = -2\) becomes \(e^x + e^{-x} = -2\). Solve it as a quadratic in \(e^x\).

Step 9: Solve the quadratic equation

Solve \((e^x)^2 + 2(e^x) - 1 = 0\) using the quadratic formula: \[e^x = \frac{-2 \pm \sqrt{4 + 4}}{2} = -1 \pm \sqrt{2}\].

Step 10: Calculate \(x\)

Only positive values for \(e^x\) are valid, so \(e^x = \sqrt{2} - 1\). Take the natural logarithm: \(x = \ln(\sqrt{2} - 1)\).

Step 11: Generalize the solution

Combine the solutions for \(z = x + iy\) such as \(z = \ln(\sqrt{2} - 1) + i\left(\frac{\pi}{2} + n\pi\right)\) for integer \(n\) since \(\cos(y) = 0\).

Key Concepts

Hyperbolic Functions

Hyperbolic functions are analogues of trigonometric functions but are derived using hyperbolas instead of circles. One of the key hyperbolic functions is the hyperbolic sine, denoted as \( \sinh z \).
This function is defined using the exponential function as \( \sinh z = \frac{e^z - e^{-z}}{2} \).
The exponential representation allows us to easily manipulate and solve equations involving \( \sinh \), particularly in complex analysis where the variable \( z \) often has a real and an imaginary component.

• Hyperbolic sine helps us model scenarios where growth is exponential, which often appears in physics and engineering.
• Unlike trigonometric functions that oscillate, hyperbolic functions grow exponentially.

In this problem, recognizing that \( \sinh z = -i \) translates into an equation involving exponential terms is crucial for finding the solution.

Exponential Form

The exponential form of complex numbers is a powerful tool that simplifies many calculations. A complex number \( z = x + iy \) can be expressed in exponential form as \( e^z = e^{x + iy} = e^x e^{iy} \).
This form separates the real and imaginary parts effectively:

• \( e^x \) is the exponential growth (or decay) component when \( x \) is real.
• \( e^{iy} \) can be expressed using Euler's formula as \( \cos(y) + i\sin(y) \).

The use of exponential form in the solution allows us to manage complex numbers and solve equations methodically.
By expressing both \( e^z \) and \( e^{-z} \) in exponential terms, we simplify our task to solving algebraic equations with real coefficients.
It's a way of transforming multiplication into addition, making things simpler.

Quadratic Equation

A quadratic equation is an equation of the form \( ax^2 + bx + c = 0 \).
Quadratics are a fundamental concept in algebra and appear frequently in mathematics.
In this exercise, solving the equation required finding the roots of a quadratic expression.
The form that appeared was \( (e^x)^2 + 2(e^x) - 1 = 0 \). By utilizing the quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), we can determine possible values for \( e^x \).
Key points in using quadratic equations include:

• Identifying coefficients \( a \), \( b \), and \( c \).
• Applying the correct order of operations to solve for \( x \).
• Selecting only realistic solutions, such as in our case, positive values of \( e^x \).

Quadratic equations often come up when two squares are involved, and understanding their solution processes is essential in both algebra and complex analysis.

Imaginary Unit

The imaginary unit, denoted as \( i \), is a fundamental component of complex numbers and is defined as \( i = \sqrt{-1} \).
This means \( i^2 = -1 \), reflecting its unique property of being a purely imaginary construct.
\( i \) allows for the expansion of real numbers into the complex plane, enabling solutions to equations that do not have real solutions otherwise.
In our hyperbolic function example, \( -i \) signifies the negative imaginary component, which shifts the problem into a complex domain.

• Complex numbers can be represented as \( a + bi \), where \( a \) is the real part and \( bi \) the imaginary part.
• The imaginary unit facilitates work with angles and oscillations in electrical engineering and wave physics.

Without \( i \), it wouldn’t be possible to fully explore and solve various phenomena or be able to understand and operate on equations where the solution involves square roots of negative numbers, as seen in this problem.