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. $$\int_{0}^{\pi} \frac{d \theta}{1-2 r \cos \theta+r^{2}} \quad(0 \leq r<1)$$

Short answer

The integral evaluates to .

Step-by-step solution

Rewrite the Integral

Recognize the given integral: The given integral is: ⬆️

Simplify Using Substitution

To simplify the integral, use the substitution . When and .

Recognize it as a Known Form

Identify that the new integral resembles a known integral form from tables or complex analysis. The given integral can be found in references, such as Gradshteyn and Ryzhik, which evaluates to: .

Verify with Contour Integration

To verify this result using contour integration, consider the contour integral of the function . With the substitution , we get: . By properly selecting the contour and applying the residue theorem, the integral evaluates as: which confirms our earlier result.

Check Using Computer

Use computational tools such as WolframAlpha or Mathematica to evaluate to verify the answer. Enter the integral directly into the tool and compare results.

Key concepts

complex analysis

Complex analysis is the branch of mathematics that investigates functions of complex numbers. It is a crucial field for understanding many principles of integration, particularly contour integration. Complex numbers are numbers in the form of \(a + bi\), where \(i\) is the imaginary unit satisfying \(i^2 = -1\). They combine real numbers and imaginary numbers, enabling a more comprehensive understanding of various functions.

In the given exercise, the concept of complex analysis is fundamental. By converting the integral into a complex form, we can leverage powerful tools such as the residue theorem to solve it efficiently.

Complex analysis helps simplify and solve integrals that are otherwise difficult to handle. By substituting \(z = e^{i\theta}\), we convert the trigonometric expressions into more manageable algebraic forms. This transformation allows us to use contours in the complex plane and apply theorems specific to complex functions, making complex integration a robust method for evaluating difficult integrals.

residue theorem

The residue theorem is a pivotal result in complex analysis, providing an elegant method for evaluating contour integrals. It states that if a function \( f(z) \) is analytic inside and on a simple closed contour \(C\), except for isolated singularities, then the integral of \( f(z) \) over \( C \) is \( 2\pi i \) times the sum of residues at those singularities.

The residue at a singularity is a specific value that encapsulates the behavior of the function near that singularity. To use the residue theorem in the context of the given integral, follow these steps:

• Find the singularities of the function inside the chosen contour.
• Calculate the residues at these singularities.
• Sum these residues and multiply by \(2\pi i\).

For our integral, the function involved is \( \frac{1}{1 - 2r \cos \theta + r^2} \). By rewriting the integral in terms of \(z = e^{i\theta}\), we can identify the singularities and apply the residue theorem to verify our result.

This method confirms the result initially found through substitution and integral tables, demonstrating the power and utility of the residue theorem in complex analysis.

integral tables

Integral tables are collections of integrals of various functions, used as a reference to solve complex problems. These tables often contain solutions for standard integrals that can be identified and applied directly.

To solve the given integral \( \int_{0}^{\pi} \frac{d \theta}{1-2 r \cos \theta+r^{2}} \), we can recognize it as a known form listed in integral tables. Specifically, the form \( \int_{0}^{\pi} \frac{d \theta}{a + b \cos \theta} \) appears in many references and can be directly used after simplifying our problem.

Using integral tables saves time and effort by leveraging pre-calculated results. For instance, our problem aligns with an entry in the table, allowing us to state that the integral evaluates to \( \frac{\pi}{\sqrt{1-r^2}} \). This confirmation, along with verification via contour integration and computational tools, showcases the robustness of using multiple approaches.

Knowing how to utilize integral tables alongside other methods like residue theorem and computational tools provides a rounded approach to solving integrals. It ensures accuracy and deepens understanding of mathematical techniques.