Problem 6 - Advanced Engineering Mathematics

Question

$$\int_{-2 i}^{1}\left(3 z^{2}-4 z+5 i\right) d z=z^{3}-2 z^{2}+\left.5 i z\right|_{-2 i} ^{1}=-19-3 i$$

Step-by-Step Solution

Answer

The given solution seems incorrect; reevaluate calculations for complex numbers.

Step 1: Identify the integral limits

In this problem, we are given the integral limits as from \(-2i\) to 1.

Step 2: Determine the antiderivative

We need to integrate the function \(3z^2 - 4z + 5i\) with respect to \(z\). The antiderivative of \(3z^2\) is \(z^3\), the antiderivative of \(-4z\) is \(-2z^2\), and the antiderivative of \(5i\) is \(5iz\). Thus, the antiderivative of the entire expression is \(z^3 - 2z^2 + 5iz\).

Step 3: Evaluate the antiderivative at the upper limit

Substitute \(z = 1\) into the antiderivative \(z^3 - 2z^2 + 5iz\) which gives us \(1^3 - 2(1)^2 + 5i(1) = 1 - 2 + 5i\). Simplifying this, we get \(-1 + 5i\).

Step 4: Evaluate the antiderivative at the lower limit

Substitute \(z = -2i\) into the antiderivative: \((-2i)^3 - 2(-2i)^2 + 5i(-2i)\). This equals to \(-8i^3 - 2(4(-1)) - 10i^2\), because \(i^2 = -1\).Therefore, it simplifies to \(-8(-i) - 8 - 10(-1) = 8i - 8 + 10\), which simplifies further to \(8i + 2\).

Step 5: Subtract the evaluations

Subtract the result from step 4 \(8i + 2\) from the result in step 3 \(-1 + 5i\):\((-1 + 5i) - (8i + 2)\).Simplifying, we obtain:\(-1 + 5i - 8i - 2 = -3 - 3i\).

Step 6: Verify the result

The calculation errors lead to \(-3 - 3i\) which contradicts the given answer \(-19 - 3i\). Rechecking: ensure correct multiplication and operations according to complex number rules before recomputing.

Key Concepts

Complex Analysis

Complex Analysis is a fascinating branch of mathematics that extends the concept of calculus to the complex numbers. It studies functions that have complex numbers as their domain and range.

• Complex analysis focuses heavily on the concept of analytic functions. These are complex functions that are differentiable in a neighborhood of every point in their domain.
• One important aspect is understanding curves and surfaces in the complex plane. These are used to represent the integration paths.
• Complex integration, as performed in the given exercise, involves integrating functions along a path in the complex plane.

Understanding complex analysis requires a good grasp of both complex numbers and calculus. The results can often be surprising and elegant, much like the completion of a definite integral using complex techniques that reveal deeper insights.

Antiderivative

The concept of an antiderivative is central to solving integrals, especially in complex analysis. An antiderivative of a function is another function whose derivative gives back the original function. In simpler terms, it is the reverse process of taking a derivative.

• For instance, the antiderivative of the polynomial function \(3z^2 - 4z + 5i\) yields \(z^3 - 2z^2 + 5iz\).
• Finding antiderivatives effectively helps in calculating indefinite or definite integrals, which is seen in this exercise.
• The process involves applying basic integration rules to find the general function that encompasses all the behavior of the original function.

Once we have the antiderivative, we can use it to evaluate the integral at given limits, determining the net effect or area under the curve in the complex plane.

Definite Integral

Definite integrals are used to compute the accumulation of values over a specified interval. In complex analysis, this interval is a path in the complex plane.

• For a definite integral, we calculate the net 'result' by evaluating the antiderivative at the upper and lower limits and subtracting the results.
• In this exercise, the definite integral ranges from \(-2i\) to 1, representing a path in the complex plane.
• Calculating this involves using the antiderivative and directly substituting these endpoints into it.

The result from a definite integral reflects the total 'change' or 'accumulation' along the defined path. The process highlights the clear difference between integrating real functions and those involving complex numbers.

Complex Numbers

Complex numbers form the backbone of complex analysis, combining real and imaginary components. In the format \(z = x + yi\), \(x\) is the real part, and \(yi\) is the imaginary part.

• Complex numbers allow for the extension of the number line to a complex plane, losing no information with real and imaginary parts both playing crucial roles.
• They have unique arithmetic operations, such as conjugation, which can be utilized in complex integration.
• When evaluating integrals like in this exercise, you must be careful with operations involving \(i\), particularly when dealing with powers, as \(i^2 = -1\).

Understanding the behavior and interaction of complex numbers is key to accurately solving problems in complex analysis and successfully performing integrations.