Question

Evaluate each of the following integrals by turning it into a contour integral around a unit circle. (a) \(\int_{0}^{2 \pi} \frac{d \theta}{5+4 \sin \theta}\). (b) \(\int_{0}^{2 \pi} \frac{d \theta}{a+\cos \theta}\) where \(a>1\). (c) \(\int_{0}^{2 \pi} \frac{d \theta}{1+\sin ^{2} \theta}\). (d) \(\int_{0}^{2 \pi} \frac{d \theta}{\left(a+b \cos ^{2} \theta\right)^{2}} \quad\) where \(a, b>0\). (e) \(\int_{0}^{2 \pi} \frac{\cos ^{2} 3 \theta}{5-4 \cos 2 \theta} d \theta\). (f) \(\int_{0}^{\pi} \frac{d \phi}{1-2 a \cos \phi+a^{2}} \quad\) where \(a \neq \pm 1\). (g) \(\int_{0}^{\pi} \frac{\cos ^{2} 3 \phi d \phi}{1-2 a \cos \phi+a^{2}} \quad\) where \(a \neq \pm 1\). (h) \(\int_{0}^{\pi} \frac{\cos 2 \phi d \phi}{1-2 a \cos \phi+a^{2}} \quad\) where \(a \neq \pm 1\). (i) \(\int_{0}^{\pi} \tan (x+i a) d x\) where \(a \in \mathbb{R}\). (j) \(\int_{0}^{\pi} e^{\cos \phi} \cos (n \phi-\sin \phi) d \phi \quad\) where \(n \in \mathbb{Z}\).

Short answer

The specific answer depends on each given integral. By turning each one into a contour integral, use the methods of complex analysis, specifically the Residue Theorem, to evaluate. The answer will be a numerical or algebraic expression in terms of the parameter included in the integral.

Step-by-step solution

Integral Conversion

For each of the given integrals, the first step is to convert the real variable into a complex variable that parametrizes a unit circle in the complex plane. To achieve this, any appearance of \(\sin \theta\) in the integrand is replaced with \((e^{i \theta} - e^{-i \theta}) / 2i\), and any appearance of \(\cos \theta\) is replaced with \((e^{i \theta} + e^{-i \theta}) / 2\). These replacements are based on the Euler's formula.

Deformation of the Contour Integral

After the conversion is performed and the integrand is completely expressed in terms of complex exponential functions, the next step is to replace the real integral over the range from 0 to \(2 \pi\) with a contour integral over the unit circle in the complex plane. In this context, differentiate \(dz = ie^{i\theta}d\theta\), and with \(d\theta = dz/ie^{i\theta}\), replace \(d\theta\) in the integral. The given limits of 0 to \(2\pi\) become a full circular path about the origin, traversed once in the counterclockwise direction.

Evaluation of the Contour Integral

Now that the contour integral is fully set up, the next step is to evaluate the contour integral. Instead of going through the process of direct integration, the calculation is based on the Residue Theorem of complex analysis, which states that the value of a contour integral of a meromorphic function over a simple closed contour is \(2\pi i\) times the sum of its residues inside the contour. Identify poles and calculate residues there to compute integral.

Key concepts

Complex Analysis

Complex analysis is a branch of mathematical analysis that deals with functions of complex numbers. Unlike real numbers, complex numbers have both a real part and an imaginary part.
They can be depicted as points or vectors in a 2-dimensional plane, often termed the complex plane. Complex analysis is powerful for solving real integrals because of its capacity to convert them into contour integrals, such as in the unit circle method outlined in our exercise.

• One key aspect of complex analysis is the study of complex functions that are differentiable, often known as holomorphic or analytic functions.
• These functions are not only differentiable in the calculus sense, but they are also infinitely differentiable, which gives them very special properties.
• The typical function considered in complex analysis is a meromorphic function, which might have isolated singularities, known as poles.

Understanding these principles is essential when working with contour integrals and applying methods like the Residue Theorem. Complex analysis provides tools and theorems, like Euler's formula, which allow us to turn trigonometric integrals into more manageable forms on the complex plane.

Residue Theorem

The Residue Theorem is a foundational theorem in complex analysis, commonly employed to evaluate contour integrals. It significantly simplifies the process by reducing the integration to algebraic calculations involving residues.
This theorem is particularly useful when dealing with contours that encircle several poles, as it allows integration over singularities within closed contours.

• The theorem states that if a function is holomorphic on a domain except for a finite number of isolated singularities, the contour integral of the function around a closed path is equal to \(2\pi i\) times the sum of the residues at each of the singularities inside the contour.
• The computation of residues involves finding the coefficients of \(\frac{1}{z-a}\) in the Laurent series expansion of the function around the pole \(a\).
• This method bypasses the often complex task of direct integration and highlights the simplicity and power of complex analysis.

Using the Residue Theorem, many seemingly intractable integrals can be tackled with efficiency, turning a complex problem into a simple summation challenge.

Contour Integrals

Contour integrals are integrals over curves in the complex plane and a central concept in complex analysis. A contour integral is defined over a path, called the contour, which may encircle one or more singularities.
They turn the integration into a geometric problem, dealing with paths taken on the complex plane.

• To convert a definite integral into a contour integral, we use a parameterization of the contour in the complex plane. For a unit circle, this usually means converting expressions using Euler's formula.
• The parameterization converts trigonometric entities into exponential forms using the relations \(\sin \theta = \frac{e^{i\theta} - e^{-i\theta}}{2i}\) and \(\cos \theta = \frac{e^{i\theta} + e^{-i\theta}}{2}\).
• In essence, contour integrals allow for a powerful method of integrating around singularities without linearizing the path around those singularities.

This transformation is key in evaluating definite integrals through methods such as those involving the Residue Theorem, and it underlies much of the theoretical power of complex analysis.

Euler's Formula

Euler's Formula is a cornerstone in complex analysis, linking the fields of algebra, geometry, and trigonometry through complex numbers. It provides the fundamental relation \(e^{i\theta} = \cos \theta + i\sin \theta\), connecting exponential functions with trigonometric functions in the context of complex numbers.
This relation is pivotal in converting real trigonometric functions into a form that can be handled using complex analysis techniques.

• This formula implies that any point on the unit circle can be represented using an exponential function with an imaginary exponent.
• Euler's formula is instrumental in parametrizing trigonometric functions in terms of complex exponentials, forming the backbone of techniques like those used in contour integration.
• It simplifies the integral conversion process by making the integrands easier to manage and often reducing them to simple algebraic forms.

In complex integration problems, Euler's Formula allows for the transformation of trigonometric integrands, enabling the application of methods like the Residue Theorem efficiently. Its elegance and utility make it an essential tool in the mathematical toolkit.