Problem 12 - Complex Analysis

Question

Show (a) $$ \int_{-\infty}^{\infty} \frac{x^{2}}{\left(x^{2}+a^{2}\right)^{2}} d x=\frac{\pi}{2 a}, \quad(a>0) $$ (b) $$ \int_{-\infty}^{\infty} \frac{d x}{\left(x^{2}+4 x+5\right)^{2}}=\frac{\pi}{2} $$ (c) \(\int_{0}^{\infty} \frac{d x}{\left(x^{2}+a^{2}\right)\left(x^{2}+b^{2}\right)}=\frac{\pi}{2 a b(a+b)}, \quad(a, b>0)\).

Step-by-Step Solution

Answer

(a) \( \frac{\pi}{2a} \); (b) \( \frac{\pi}{2} \); (c) \( \frac{\pi}{2ab(a+b)} \)."}

Step 1: Problem (a) – Identify the integral and use substitution

To solve \( \int_{-\infty}^{\infty} \frac{x^{2}}{(x^{2}+a^{2})^{2}} \, dx \), consider the substitution \( u = \frac{x}{a} \), so \( du = \frac{dx}{a} \) and \( x = au \). The integral becomes \[ a \int_{-\infty}^{\infty} \frac{a^2u^2}{(a^2u^2+a^2)^2} \, du = a \int_{-\infty}^{\infty} \frac{u^2}{(u^2+1)^2} \, du. \]

Step 2: Problem (a) – Evaluate the new integral

We can evaluate \( \int_{-\infty}^{\infty} \frac{u^2}{(u^2+1)^2} \, du \) using residues. Consider the complex integral \( \int_{\Gamma} \frac{z^2}{(z^2+1)^2} \, dz \) where \( \Gamma \) is a semi-circular contour in the upper half-plane. The poles are at \( z = i \). Compute the residue at \( z = i \), which after simplification gives \( \frac{\pi}{2} \). Then multiply by \( a \) to find the value of the original integral \( \int_{-\infty}^{\infty} \frac{x^2}{(x^2+a^2)^2} \, dx = \frac{\pi}{2a} \).

Step 3: Problem (b) – Complete the square and identify integral form

For \( \int_{-\infty}^{\infty} \frac{dx}{(x^2+4x+5)^2} \), write the expression as \((x+2)^2 + 1 \) by completing the square. The integral now becomes \( \int_{-\infty}^{\infty} \frac{dx}{((x+2)^2+1)^2} \).

Step 4: Problem (b) – Use known integral result

The integral \( \int_{-\infty}^{\infty} \frac{1}{(x^2+1)^2} \, dx = \frac{\pi}{2} \). By making the substitution \( u = x + 2 \), the integral remains the same type, leading to the result \( \int_{-\infty}^{\infty} \frac{dx}{((x+2)^2+1)^2} = \frac{\pi}{2} \).

Step 5: Problem (c) – Decompose into partial fractions

Consider \( \int_{0}^{\infty} \frac{dx}{(x^2+a^2)(x^2+b^2)} \). Express the integrand as \( \frac{A}{x^2+a^2} + \frac{B}{x^2+b^2} \) using partial fraction decomposition. Solve the equation by equating coefficients.

Step 6: Problem (c) – Evaluate each partial fraction integral

Integrate each partial fraction separately by using \( \int \frac{dx}{x^2+a^2} = \frac{\pi}{2a} \). Combine the results using the partial fraction decomposition's coefficients, leading to \( \frac{\pi}{2ab(a+b)} \).

Key Concepts

Integral Calculus

Integral calculus is a branch of mathematics focused on finding antiderivatives and calculating the area under curves. It deals with integrals, which are essential for solving problems related to accumulation. In the provided exercise, the focus is on evaluating definite integrals, which involve specific bounds.

• A definite integral computes the net area between a function and the x-axis over an interval.
• In our context, the goal is to evaluate integrals with infinite limits, extending from negative to positive infinity.
• Integration techniques like substitution help simplify the process.

For example, when performing substitution, we change the variable of integration to simplify the integral into a more manageable form. Once simplified, evaluation becomes straightforward, especially when using further complex analysis techniques. Calculus is central to understanding change, making it a foundational tool in mathematics.

Contour Integration

Contour integration is a method in complex analysis, used particularly for evaluating complex-valued integrals along specified paths in the complex plane. It’s a powerful tool, especially for solving integrals over curves defined on this plane.

• A contour is any path or curve in the complex plane. These might include straight lines, circles, or combinations of different types of paths.
• The method uses path integrals where the function is integrated along these contours.
• In the exercise, a semicircular contour in the upper half-plane is employed to evaluate the integral from negative to positive infinity.

This technique is beneficial because it transforms complex contour integrals into simpler real line integrals. It utilizes the properties of holomorphic functions and assists in addressing the whole-domain behavior of functions, simplifying the evaluation process tremendously.

Residue Theorem

The residue theorem is a pivotal concept in complex analysis, allowing for the evaluation of line integrals of analytic functions over closed curves using residues. This theorem extends the practice of contour integration.

• The residue of a function at a pole is essentially the coefficient of \( (z - a)^{-1} \) in its Laurent series expansion around that point.
• The contour integral of a function over a closed path equals \( 2\pi i \) times the sum of the residues inside the contour.
• In our example, the integral is solved by focusing on the pole at \( z = i \), within the chosen contour.

Using the residue theorem allows us to quickly tackle seemingly complex integrals by reducing the problem to calculating residues. This characteristic is pivotal in simplifying and solving real-integral problems related to infinite limits.

Partial Fraction Decomposition

Partial fraction decomposition is a technique often used in integral calculus to simplify integrands. It expresses a complex rational function as the sum of simpler, fraction components.

• It’s especially useful when dealing with integrands consisting of polynomial ratios.
• The method involves breaking down the function into a sum of terms whose integrals are easier to manage.
• In exercise part (c), the decomposition separates \( \frac{1}{(x^2+a^2)(x^2+b^2)} \) into simpler fractions.

Each simpler fraction in the decomposition can be integrated individually, often using known integral results. This technique can handle complex polynomial denominators by making the integration more straightforward and is a crucial skill when working with calculus problems.