Problem 12 - Complex Analysis for Mathematics and Engineering

Question

Use residues to find $$ \int_{0}^{2 \pi} \frac{1}{\left(1+8 \cos ^{2} \theta\right)^{2}} d \theta $$

Step-by-Step Solution

Answer

The integral evaluates to \( \frac{2\pi}{27} \).

Step 1: Convert the Integral to Complex Form

Use the identity \( e^{i\theta} = \cos\theta + i\sin\theta \) to express the integral in terms of a complex variable. Set \( z = e^{i\theta} \), then \( dz = i e^{i\theta} d\theta = iz d\theta \), or \( d\theta = \frac{dz}{iz} \). The cosine function becomes \( \cos\theta = \frac{z + \frac{1}{z}}{2} \), so \( \cos^2\theta = \frac{(z + \frac{1}{z})^2}{4} = \frac{z^2 + 2 + \frac{1}{z^2}}{4} \). Rewrite the integral:\[ \begin{aligned} \int_{0}^{2\pi} \frac{1}{(1 + 8\cos^2\theta)^2} d\theta &= \oint_{|z|=1} \frac{1}{(1 + 8 \left( \frac{z^2 + 2 + \frac{1}{z^2}}{4} \right) )^2} \frac{dz}{iz}. \end{aligned} \]

Step 2: Simplify the Integrand

Substitute \( \cos^2\theta = \frac{z^2 + 2 + \frac{1}{z^2}}{4} \) into the expression:\[ 8\cos^2\theta = 2(z^2 + 2 + \frac{1}{z^2}) = 2z^2 + 4 + \frac{2}{z^2}. \]Thus, the denominator becomes:\[ \left(1 + 8 \cos^2\theta\right)^2 = \left(1 + 2z^2 + 4 + \frac{2}{z^2} \right)^2 = \left(5 + 2z^2 + \frac{2}{z^2} \right)^2. \]Then rewrite the integrand based on this simplification.

Step 3: Find the Poles

The expression now has the form:\[ \frac{1}{(5 + 2z^2 + \frac{2}{z^2})^2} \].To find poles, set the denominator equal to zero and solve the equation \( 5 + 2z^2 + \frac{2}{z^2} = 0 \). This equation simplifies to \( 2z^4 + 5z^2 + 2 = 0 \). Solve for \( z^2 \) and, subsequently, \( z \) using the quadratic formula. The roots give the location of poles inside the unit circle.

Step 4: Apply Residue Theorem

For each pole found in the previous step that lies inside the unit circle, calculate the residue of the function. The integrand is now \( \frac{1}{iz (Q(z))^2} \) where \( Q(z) = 5 + 2z^2 + \frac{2}{z^2} \). Calculate the residue for each pole by first differentiating \( Q(z) \). Then apply the Residue Theorem:\[ \oint_{|z|=1} f(z) \, dz = 2\pi i \sum \text{Residues inside } |z|=1. \]

Step 5: Compute the Desired Integral

Sum the residues found in Step 4 and multiply by \( 2\pi i \) to find the result of the integral. Simplify the expression to get the numerical value of the original real integral:\( \int_{0}^{2\pi} \frac{1}{(1+8\cos^2\theta)^2} \, d\theta = \text{Real part of } (2\pi i \sum \text{Residues}). \) Compute to find the answer.

Key Concepts

Residue Theorem

The Residue Theorem is a powerful tool in complex analysis, particularly useful for evaluating complex integrals. It relates the contour integration of a holomorphic function around a closed loop to the sum of residues of the function inside the loop. This theorem states that for a function "f" holomorphic on the interior and exterior of a contour "C" except at isolated singularities,

• \( \oint_{C} f(z) \, dz = 2 \pi i \sum \text{Residues of } f(z) \text{ inside } C \)

This simplifies complex integrations by focusing on poles, where residues represent the contribution of each pole to the integral. Applying this to the given exercise, the integral over \( [0, 2\pi] \) is converted to a contour integral over the unit circle, simplifying the problem significantly. Once the residues are computed, the integral is easily solved using this powerful theorem.

Contour Integration

Contour integration involves calculating the integral of a complex function along a contour or path in the complex plane. It is a conceptual shift from real analysis, considering paths in two dimensions with real and imaginary parts.
In the context of the exercise, the real integral transforms into an integral over the unit circle in the complex plane by changing variables to \( z = e^{i\theta} \). This change frames the problem in terms of complex numbers and simplifies the integration because the path in complex analysis captures essential information about the function via its residues.

• The contour here is the unit circle, typically denoted \( |z| = 1 \).
• The function defined over this contour could have singularities—points where it tends to infinity.

Once the transformation is complete, contour integration allows us to handle these singularities with methods such as the Residue Theorem, focusing only on the contributions from poles inside the contour.

Complex Numbers

Complex numbers are a central element in the study of complex analysis, significantly expanding the real number system. A complex number is of the form \( z = x + yi \), where \( x \) and \( y \) are real numbers, and \( i \) is the imaginary unit satisfying \( i^2 = -1 \).

• The real part, \( x \), and the imaginary part, \( y \), allow complex numbers to form a plane—often called the complex plane.
• Complex arithmetic involves operations such as addition, multiplication, and conjugation, which play crucial roles in simplifying expressions and solving equations.

In our exercise, the conversion from \( \theta \) to a complex representation \( z = e^{i\theta} \) exemplifies the elegance of complex numbers. This formulation expresses trigonometric functions like \( \cos\theta \) in terms of \( z \), bridging real analysis and complex analysis, providing simplifications that are not as readily apparent in the realm of real numbers alone.