Question

The values of the following integrals are known and can be found in integral tables or by computer. Your goal in evaluating them is to learn about contour integration by applying the methods discussed in the examples above. Then check your answers by computer. $$\int_{0}^{\infty} \frac{x^{2} d x}{x^{4}+16}$$

Short answer

The value of the integral \( \int_{0}^{\infty} \frac{x^{2} d x}{x^{4}+16} \) is \ \frac{\pi}{8} \.

Step-by-step solution

Recognize the Form of the Integral

Identify that the integral \(\( \int_{0}^{\infty} \frac{x^{2} d x}{x^{4}+16} \)\) resembles the form of integrals that can be evaluated using contour integration methods with complex analysis.

Rewrite the Polynomial in the Denominator

Notice that \(x^{4} + 16 \) can be factored as \((x^2 + 2i)(x^2 - 2i)\). This helps in identifying the appropriate contours and residues for the integration.

Use the Residue Theorem

Set up the integral in the complex plane using contour integration techniques. Identify the poles of the function \( \frac{z^{2}}{z^{4}+16} \) and use the residue theorem to evaluate.

Find the Poles and Residues

The poles are solutions to \(z^{4} + 16 = 0\), which are \( z = \pm2^{1/2} e^{i\pi/4} \) and \( z = \pm2^{1/2} e^{-i\pi/4} \). Calculate the residues at these poles.

Evaluate the Integral Using Residues

Sum up the residues inside the upper half-plane contour to find the value of the integral. Each residue at \( 2^{1/2} e^{i\pi/4} \) and \( 2^{1/2} e^{-i\pi/4} \) contributes to the total.

Verify the Solution

Check the computed result using computational tools or integral tables to ensure correctness.

Key concepts

residue theorem

The residue theorem is a powerful tool in complex analysis. It is used to evaluate complex integrals, particularly those involving meromorphic functions, by summing the residues of the integrand at its poles. Here's a breakdown of the key aspects:

• The residue theorem states that if a function is analytic inside and on some closed contour, except for a finite number of singular points (poles), the integral of the function around the contour is \(2\,\pi\,i\) times the sum of the residues at the poles inside the contour.
• To apply the residue theorem, you need to identify the poles of the function within the chosen contour and calculate the residues at those poles.
• The residue at a pole is the coefficient of the \((z - z_0)^{-1}\) term in the Laurent series expansion of the function around the pole \(z_0\).

In practical contexts, such as the integral problem given, this means writing the integrand in a form that reveals its poles and then using the residue theorem to sum the results of these particular points.

complex analysis

Complex analysis is a branch of mathematics that deals with functions of complex variables. It's a powerful field with applications across various domains such as engineering, physics, and number theory. Some fundamental ideas include:

• **Complex Numbers:** These are numbers in the form \(z = x + yi\), where \(x\) and \(y\) are real numbers and \(i\) is the imaginary unit, satisfying \(i^2 = -1\).
• **Analytic Functions:** Functions that are complex differentiable in a neighborhood of every point in their domain. Examples are polynomials and exponential functions.
• **Contours and Path Integrals:** Contours are paths in the complex plane, and a contour integral involves integrating a complex function along a given path. This provides a framework for solving complex integrals using techniques like the residue theorem.

In the context of the previous integral problem, we used complex analysis to rewrite the integrand, identify its poles, and apply the residue theorem within an appropriate contour to evaluate the integral.

integral evaluation

Integral evaluation, particularly in the context of contour integration and complex analysis, involves several key steps:

• **Identifying the Form of the Integral:** Recognizing integrals that can benefit from complex analysis methods, such as those involving rational functions.
• **Rewriting Polynomials:** Breaking down the polynomial in the denominator into factors to better identify poles and simplify the integrand.
• **Set Up in Complex Plane:** Transforming the real integral into an equivalent complex integral by choosing a contour in the complex plane that simplifies the evaluation process.
• **Finding Poles and Residues:** Locating the singularities (poles) of the integrand within the chosen contour and calculating the residues at those points.
• **Applying the Residue Theorem:** Summing the residues inside the contour and multiplying by \(2\pi i\) to find the value of the integral.

In the provided example, this process involved factoring the denominator, setting up the contour, identifying the poles \(z = \pm 2^{1/2} e^{i\pi/4}\) and \(z = \pm 2^{1/2} e^{-i\pi/4}\), and finally summing the residues.