Question

Evaluate \(\oint_{C} d z /\left(z^{2}-1\right)\) where \(C\) is the circle \(|z|=3\) integrated in the positive sense. Hint: Deform \(C\) into a contour \(C^{\prime}\) that avoids the singularities of the integrand. Then use Cauchy Goursat theorem.

Short answer

To find the value of the contour integral \(\oint_{C} dz / (z^2 -1)\), one needs to compute the negative sum of the integrals over two circular sectors \(C_2\) and \(C_3\) that form, together with a circle of radius two, a deformed contour \(C'\) that avoids the singularities of the integrand. The integral over \(C'\) itself is zero by the Cauchy-Goursat theorem, which implies the relation between the integrals over the original and deformed contours.

Step-by-step solution

Identifying Singularities

First it's important to recognise where the singularities of the function are. The function \(f(z) = 1 / (z^2 - 1)\) has singularities at \(z = 1\) and \(z = -1\).

Deform the Contour

The initial contour \(C\) is a circle of radius 3, which includes both the singularities. Given that we can deform the contour as long as it does not cross the singularities, we can deform \(C\) into a new contour \(C'\) consisting of a circle of radius two, \(C_1\), and two circular sectors, \(C_2\) and \(C_3\), with respective radii of 1.

Apply Cauchy-Goursat Theorem

The Cauchy-Goursat is a theorem from complex analysis that allows us to consider the integral of this slightly larger contour \(C'\). Since the integrand \(f\) is a holomorphic function everywhere except the singularities, which are now avoided by \(C'\), we can apply the Cauchy-Goursat theorem. According to this theorem, the contour integral of a holomorphic function over any closed contour is zero. Hence, the integral \(\oint_{C'} dz / (z^2 -1)\) is equal to zero.

Calculate Integral over Deformed Contour

Since the contour \(C'\) is made up of \(C_1\), \(C_2\), and \(C_3\), the integral over \(C'\) is the sum of the integrals over these parts: \(\oint_{C'} dz / (z^2 -1) = \oint_{C1} dz / (z^2 -1) + \oint_{C2} dz / (z^2 -1) + \oint_{C3} dz / (z^2 -1)\). But because of the relation between the integrals over \(C'\) and \(C\) and the Cauchy-Goursat theorem, we find that \(\oint_{C} dz / (z^2 -1) = - \oint_{C2} dz / (z^2 -1) - \oint_{C3} dz / (z^2 -1)\), i.e., it is equal to the negative sum of the integrals over \(C_2\) and \(C_3\). This is the quantity we need to compute to find the desired integral.

Key concepts

Complex Analysis

Complex analysis is a fascinating and rich area of mathematics, focusing on the study of complex numbers and functions of a complex variable. A complex number has a real part and an imaginary part and is typically expressed in the form of a + bi, where a and b are real numbers, and i is the imaginary unit with the property that i^2 = -1. In complex analysis, we delve into the behavior of functions when their inputs are these complex numbers.

One of the fundamental results in complex analysis is that functions which are differentiable in terms of complex variables often possess remarkable properties like infinite differentiability, and they can be represented by power series within their domain of differentiability, known as holomorphic functions. Holomorphic functions are the core subject of complex analysis, as they exhibit the features that make this field of mathematics truly powerful, like conformality and the ability to evaluate integrals using the residues of functions at their singularities.

Contour Integration

Contour integration is a method of integrating complex functions along a path or contour in the complex plane. It's a crucial concept in complex analysis and has applications ranging from theoretical physics to engineering. The contour can be any piecewise-smooth path, and the integral is evaluated as the sum of the function's values along this path.

To perform contour integration, we often parametrize the path by a continuous function that maps an interval of real numbers into the complex plane. Then, the complex integral can be computed as an ordinary real integral over this interval. Contour integration becomes incredibly powerful when combined with the various theorems of complex analysis, such as Cauchy's Integral Theorem, which states that the integral around a closed contour is zero provided the function is holomorphic within the region enclosed by the contour, and Cauchy's Integral Formula, which allows us to compute integrals of holomorphic functions easily.

Singularities in Complex Functions

Singularities in complex functions are points where a function does not behave well, either becoming unbounded or not well-defined. The simplest types of singularities are called poles, and others include essential singularities and branch points. For a function to be integrated smoothly over a contour, it must be free of singularities within the contour or on its path.

An important aspect of complex analysis is dealing with these singularities carefully, especially when performing contour integration. The Cauchy-Goursat theorem used in contour integration assumes the absence of singularities in the region bounded by the contour. If there are singularities, they must be handled individually, often by deforming the contour to exclude them, as mentioned in the exercise solution, or by calculating their residues if one aims to find the value of a principal value integral.

Holomorphic Functions

Holomorphic functions, also known as analytic functions, are the superheroes of complex analysis. These functions are differentiable everywhere within their domain, which means their derivative is well-defined and unique at all points. Furthermore, holomorphic functions have a power series expansion at any point in their domain, much like Taylor series for real functions.

To qualify as holomorphic, a function must satisfy the Cauchy-Riemann equations, which are a set of two real partial differential equations. The importance of holomorphic functions lies in their predictability and the ability to manipulate them using complex analysis tools like power series, residues, and various powerful integration theorems. In the context of the exercise solution provided, we consider the integrand 1 / (z^2 - 1) as a function that is holomorphic everywhere except at its singularities at z = 1 and z = -1, allowing us to use the Cauchy-Goursat theorem on an appropriately deformed contour that avoids these points.