Problem 17 - Advanced Engineering Mathematics

Question

$$\begin{aligned} \int_{i}^{1+\frac{\pi}{2} i} \sinh 3 z d z &=\left.\frac{1}{3} \cosh 3 z\right|_{i} ^{1+\frac{\pi}{2} i}=\frac{1}{3}\left[\cosh \left(3+\frac{3 \pi}{2} i\right)-\cosh 3 i\right] \\ &=\frac{1}{3}\left[\cosh 3 \cos \frac{3 \pi}{2}+i \sinh 3 \sin \frac{3 \pi}{2}-\cos 3\right]=-\frac{1}{3} \cos 3-\frac{1}{3} i \sinh 3=0.3300-3.3393 i \end{aligned}$$

Step-by-Step Solution

Answer

The integral evaluates to approximately \( 0.3300 - 3.3393i \).

Step 1: Understanding the Problem

We need to evaluate the integral \( \int_{i}^{1+\frac{\pi}{2} i} \sinh 3z \ dz \). This involves computing the antiderivative of the integrand, applying the Fundamental Theorem of Calculus, and evaluating the definite integral from \(i\) to \(1+\frac{\pi}{2} i\).

Step 2: Find the Antiderivative

The antiderivative of \( \sinh 3z \) is \( \frac{1}{3} \cosh 3z \). This comes from the property of hyperbolic functions that the derivative of \( \cosh z \) is \( \sinh z \).

Step 3: Apply Limits using the Fundamental Theorem of Calculus

According to the Fundamental Theorem of Calculus, we evaluate the antiderivative at the upper and lower bounds of the integral: \( \left.\frac{1}{3} \cosh 3z\right|_{i}^{1+\frac{\pi}{2} i} = \frac{1}{3} [ \cosh(3(1+\frac{\pi}{2} i)) - \cosh(3i) ] \).

Step 4: Evaluate \( \cosh 3z \) at the Upper Bound

Evaluate \( \cosh(3 + \frac{3\pi}{2} i) \), which involves using Euler's formula. We have \( \cosh(3 + \frac{3\pi}{2} i) = \cosh 3 \cos \frac{3\pi}{2} + i \sinh 3 \sin \frac{3\pi}{2} \). Here, \( \cos \frac{3\pi}{2} = 0 \) and \( \sin \frac{3\pi}{2} = -1 \). Thus, the expression simplifies to \( -i \sinh 3 \).

Step 5: Evaluate \( \cosh 3z \) at the Lower Bound

Now evaluate \( \cosh(3i) \) using the identity \( \cosh(zi) = \cos z \). This gives us \( \cos 3 \).

Step 6: Combine Results

Substitute the evaluated expressions back: \( \frac{1}{3} [-i \sinh 3 - \cos 3] = -\frac{1}{3} i \sinh 3 - \frac{1}{3} \cos 3 \).

Step 7: Simplify the Expression

Calculate numerical values: \( \sinh 3 \approx 10.0179 \), resulting in \( -\frac{1}{3} i \times 10.0179 \approx -3.3393i \) and \( \cos 3 \approx -0.98999 \), resulting in \( -\frac{1}{3} \times -0.98999 \approx 0.3300 \). Thus, the final result is \( 0.3300 - 3.3393i \).

Key Concepts

Hyperbolic Functions

Hyperbolic functions are analogous to trigonometric functions but are based on hyperbolas instead of circles. They are defined using exponential functions and have many useful properties in mathematics.

The primary hyperbolic functions are the hyperbolic sine denoted as \( \sinh z \) and the hyperbolic cosine denoted as \( \cosh z \). These functions are defined as follows:

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

It's important to notice their derivatives:

• The derivative of \( \sinh z \) is \( \cosh z \).
• The derivative of \( \cosh z \) is \( \sinh z \).

Hyperbolic functions are used extensively in calculus and complex analysis because they relate to exponential growth and decay models.

In complex integration, as seen in our exercise, hyperbolic functions help solve integrals involving complex variables by transforming them into more manageable expressions.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus connects differentiation with integration, showing that they are essentially inverse processes. This theorem has two main parts:

1. The first part states that if a function is continuous over an interval, an antiderivative can be found through integration.

2. The second part, which we use here, says that if you have a continuous function on \([a, b]\) and its antiderivative \(F(x)\), then:

• \( \int_a^b f(x) \, dx = F(b) - F(a) \)

This means you can evaluate definite integrals by finding antiderivatives and then applying the limits.

In the given exercise, this theorem was applied to calculate \( \int_{i}^{1+\frac{\pi}{2} i} \sinh 3z \, dz \) by using the antiderivative \( \frac{1}{3} \cosh 3z \) and then evaluating it at the given limits. These calculated boundary values allowed us to find the solution \(0.3300 - 3.3393i\).

Euler's Formula

Euler's formula is a powerful tool in complex analysis that relates complex exponentials to trigonometric functions. It states:

• \( e^{ix} = \cos x + i \sin x \)

This elegant formula provides a bridge between exponential functions and sinusoidal functions, making it essential in fields like signal processing and physics.

In our integral solution, we apply Euler's formula to evaluate expressions like \( \cosh(3 + \frac{3\pi}{2} i) \). Using the trigonometric identities derived from Euler's formula, such as \( \sin \frac{3\pi}{2} = -1 \), simplifies the process of finding the values of complex expressions.

Euler's formula is not only vital for simplifying calculations involving complex numbers, it deepens our understanding of the inherent beauty and interconnectedness of mathematical functions. So, when dealing with hyperbolic functions within complex integration, Euler's formula turns an intimidating task into a more approachable one.