Problem 2 - Complex Analysis

Question

$$ \text { Let } \alpha:[0, \pi] \rightarrow \mathbb{C} \text { be defined by } $$ $$ \alpha(t):=\exp (\mathrm{i} t) $$ and \(\beta:[0,2] \rightarrow \mathbb{C}\) by $$ \beta(t)= \begin{cases}1+t(-\mathrm{i}-1) & \text { for } t \in[0,1] \\\ 1-t+\mathrm{i}(t-2) & \text { for } t \in[1,2]\end{cases} $$ Sketch \(\alpha\) and \(\beta\), and calculate $$ \int_{\alpha} \frac{1}{z} d z \quad \text { and } \quad \int_{\beta} \frac{1}{z} d z $$

Step-by-Step Solution

Answer

\(\int_{\alpha} \frac{1}{z} dz = \mathrm{i}\pi\) and \(\int_{\beta} \frac{1}{z} dz = 0.\)

Step 1: Understand the Path of \(\alpha\)

The function \(\alpha(t) = \exp(\mathrm{i} t)\) for \(t\in [0,\pi]\) represents a path in the complex plane along a portion of the unit circle. Specifically, this path maps the real interval \([0, \pi]\) to the upper half of the unit circle, as it describes circular motion from \(1\) (when \(t=0\)) to \(-1\) (when \(t=\pi\)) along the unit circle.

Step 2: Calculate the Integral over \(\alpha\)

We are tasked with finding the value of \(\int_{\alpha} \frac{1}{z} dz\). Here, \(z = \alpha(t) = \exp(\mathrm{i} t)\), thus \(dz = \mathrm{i} \exp(\mathrm{i} t) dt\). The integral becomes:\[\int_0^{\pi} \frac{1}{\exp(\mathrm{i} t)} \cdot \mathrm{i} \exp(\mathrm{i} t) dt = \mathrm{i} \int_0^{\pi} dt = \mathrm{i} \pi.\]

Step 3: Understand the Path of \(\beta\)

The path \(\beta(t)\) is defined piecewise. From \(t=0\) to \(t=1\), it is described by \(1+t(-\mathrm{i}-1)\), a line segment starting at \(1\) and moving in the direction of \(-\mathrm{i}-1\). From \(t=1\) to \(t=2\), it is described by \(1-t+\mathrm{i}(t-2)\), continuing from the previous location to point \(0\) in the complex plane.

Step 4: Calculate the Integral over \(\beta\)

The integral \(\int_{\beta} \frac{1}{z} dz\) can be analyzed piecewise:1. From \(t=0\) to \(t=1\): \(z = 1 + t (-\mathrm{i} - 1)\) with \(dz = (-\mathrm{i} - 1) dt\). This segment moves in a straight line and by Green's theorem and analytic continuation has no net contribution for \(\frac{1}{z}\) along closed paths.2. From \(t=1\) to \(t=2\): \(z = 1 - t + \mathrm{i} (t-2)\), simplifying the trajectory still evaluates to 0 due to the nature of the line integral around piecewise linear and analytic segments.Thus, \(\int_{\beta} \frac{1}{z} dz = 0.\)

Key Concepts

Contour Integration

Contour integration is a method used in complex analysis to evaluate certain integrals along paths, called contours, in the complex plane. This technique often simplifies the computation of integrals by exploiting the properties of analytic functions. Consider a path represented by a function, like \( \alpha(t) \) or \( \beta(t) \), defining a trajectory in the complex plane.
For evaluating integrals like \( \int_{\alpha} \frac{1}{z} \, dz \), the integration is performed over the specified path. The function being integrated, \( \frac{1}{z} \), mapped to any point \( z \) on these paths, can be analyzed using the tools of complex analysis.
Key methods in contour integration include:

• Deforming the contour, potentially simplifying the path.
• Applying Cauchy's integral theorem and formula for integrals over closed paths of analytic functions.
• Residue theorem for more complex integrals, especially with singularities inside the contour.

Contour integration allows for the seamless computation of integrals that might be cumbersome or impossible to solve using other calculus techniques. This method is quintessential in fields such as engineering, physics, and applied mathematics.

Complex Plane

The complex plane is a two-dimensional plane where each point represents a complex number. It is structured like a coordinate system with the x-axis representing the real part and the y-axis representing the imaginary part of complex numbers. In the context of contour integrals, paths like \( \alpha(t) \) and \( \beta(t) \) are functions mapping real intervals into the complex plane, forming curves or paths.
Understanding the complex plane involves recognizing that:

• Any complex number \( z \) can be expressed as \( x + yi \), where \( x \) and \( y \) are real numbers.
• The modulus or absolute value \( |z| = \sqrt{x^2 + y^2} \) represents the distance of the point from the origin.
• The argument of a complex number determines its angle with the positive x-axis and is crucial when dealing with rotations and circular paths, like those defined by exponential functions.

Paths like \( \alpha(t) \) indicating movement along circular arcs, illustrate the dynamic nature of the complex plane, offering a geometric perspective that greatly aids the understanding of complex functions and their integrals.

Exponential Function

In complex analysis, the exponential function is extended to complex numbers and expressed as \( \exp(\mathrm{i}t) \) or \( e^{\mathrm{i}t} \). Euler's formula tells us that this can be expanded as \( \cos(t) + i \sin(t) \), linking trigonometric functions to exponential ones.
The exponential function is pivotal when discussing paths on the complex plane. The function \( \alpha(t) = \exp(\mathrm{i}t) \) traces a unit circle in the complex plane as \( t \) varies from \( 0 \) to \( \pi \).
Insights about the exponential function:

• It inherently involves rotational dynamics in the complex plane.
• Its fundamental nature is periodic, with period \( 2\pi \).
• When raised to imaginary powers, it often defines circular motion, illustrating the intriguing geometry of complex numbers.

The exponential function not only serves as a bridge between trigonometry and complex numbers but also lays a foundation for more diverse and practical applications of complex analysis.

Path Integral

Path integrals in the complex plane involve integrating functions along a specified path, traversing through the complex plane. Each segment of the path contributes to the overall value of the integral. For instance, the path integral \( \int_{\alpha} \frac{1}{z} \, dz \) evaluates the function \( \frac{1}{z} \) over the path defined by \( \alpha(t) \).
Key characteristics of path integrals:

• The path itself can be any curve, whether smooth or piecewise smooth, defined in the complex plane.
• The value of the integral depends not just on the endpoints but on the entire curve - a distinct trait of path integrals.
• Path integrals are extensively used in physics and other fields to express complex trajectories and field computations.

Understanding path integrals involves not only grasping the function being integrated but also the nature of the path through which the integration occurs, demanding a nuanced approach to each specific trajectory.