Problem 19 - Advanced Engineering Mathematics

Question

$$\int_{-4 i}^{4 i} \frac{1}{z^{2}} d z=-\left.\frac{1}{z}\right|_{-4 i} ^{4 i}=-\left[\frac{1}{4 i}-\left(\frac{1}{-4 i}\right)\right]=\frac{1}{2} i$$

Step-by-Step Solution

Answer

The integral evaluates to \( \frac{i}{2} \).

Step 1: Understanding the Function and Limits

We need to compute the integral of the function \( f(z) = \frac{1}{z^2} \) over the limits from \( -4i \) to \( 4i \). This function represents a simple rational function of \( z \).

Step 2: Integration of \( \frac{1}{z^2} \)

To integrate \( \frac{1}{z^2} \), we use the formula for the antiderivative: if \( f(z) = z^n \) where \( n eq -1 \), then its antiderivative is \( \frac{z^{n+1}}{n+1} \). Here \( n = -2 \), so the antiderivative is \( \frac{-1}{z} \).

Step 3: Substituting the Limits into the Antiderivative

We substitute the upper and lower limits \( 4i \) and \( -4i \) into the antiderivative: \(-\left.\frac{1}{z}\right|_{-4i}^{4i}\). This gives us: \(-\left[\frac{1}{4i} - \frac{1}{-4i}\right]\).

Step 4: Simplifying the Expression

Simplify \(-\left[\frac{1}{4i} - \frac{1}{-4i}\right]\):1. Compute \( \frac{1}{4i} \): multiply numerator and denominator by \( i \) to get \( -\frac{i}{4} \).2. Compute \( \frac{1}{-4i} \): similarly multiply to get \( \frac{i}{4} \).3. The expression becomes: \(-\left( -\frac{i}{4} - \frac{i}{4} \right) = -\left( -\frac{i}{2} \right) = \frac{i}{2}.\)

Step 5: Final Answer

After simplification, we find that the integral \( \int_{-4i}^{4i} \frac{1}{z^2} dz \) evaluates to \( \frac{i}{2} \).

Key Concepts

Antiderivative

In calculus, an antiderivative refers to a function whose derivative is the given function. When you are looking for an antiderivative, you are essentially performing the reverse operation of differentiation. This is a significant concept in integral calculus.

• To find the antiderivative, we seek a function whose rate of change (or derivative) matches the original function.
• For a function in the form of \( f(z) = z^n \) where \( n eq -1 \), the antiderivative is \( \frac{z^{n+1}}{n+1} \).

In the case of \( f(z) = \frac{1}{z^2} \), we rewrite it as \( z^{-2} \). Applying the formula gives us \( -\frac{1}{z} \) as the antiderivative. Finding an antiderivative is the first step towards computing an integral.

Rational Function

A rational function is defined as a ratio of two polynomials. In mathematical terms, it looks like \( f(z) = \frac{P(z)}{Q(z)} \), where both \( P(z) \) and \( Q(z) \) are polynomial functions.

• These functions can often be broken down into simpler parts, which can be easier to analyze or integrate.
• Rational functions are particularly interesting in complex analysis due to their behavior near poles and zeros.

For the exercise at hand, \( \frac{1}{z^2} \) is our rational function, where the polynomial in the numerator is a constant (1), and the polynomial in the denominator is \( z^2 \). Recognizing this helps us determine the nature of the function and how to handle its integration.

Complex Numbers

Complex numbers extend the idea of the one-dimensional number line to a two-dimensional complex plane using an imaginary unit, \( i \). A complex number is expressed as \( z = a + bi \), where \( a \) and \( b \) are real numbers.

• The real part, \( a \), and the imaginary part, \( bi \), are crucial for operations involving complex numbers.
• In integration involving complex numbers, the limits themselves can be complex, such as from \( -4i \) to \( 4i \).

When integrating complex functions like \( \frac{1}{z^2} \), we treat the complex number \( i \) similarly to a real number, maintaining rigor in algebraic manipulation. This understanding allows us to interpret complex integrals in the complex plane.

Definite Integral

A definite integral represents the signed area under a curve defined by a function over a specific interval. The process involves evaluating the antiderivative at specified bounds and subtracting accordingly.

• The definite integral has limits which, in the exercise, range from \( -4i \) to \( 4i \).
• This provides a numerical value representing the accumulation of quantities expressed by the function from one point to another on the complex plane.

To solve a definite integral:1. Find the antiderivative of the function.2. Substitute the upper and lower bounds into the antiderivative.3. Subtract the evaluated value at the lower limit from the evaluated value at the upper limit.For example, after integrating the function, we substitute the complex limits and perform the necessary arithmetic to achieve the result, \( \frac{1}{2} i \), as shown in the exercise. This represents the net accumulation in moving from \( -4i \) to \( 4i \) along the complex plane.