Problem 1 - Complex Analysis for Mathematics and Engineering

Question

Find a parametrization of the line that (a) joins the origin to the point \(1+i\). (b) joins the point. 1 to the point \(1+i\). (c) joins the point \(i\) to the point \(1+i\). (d) joins the point 2 to the point \(1+i\).

Step-by-Step Solution

Answer

(a) \( z(t) = t(1+i) \); (b) \( z(t) = 1 + ti \); (c) \( z(t) = i + t(1-t) \); (d) \( z(t) = 2 - t + ti \).

Step 1: Understanding Parametrization of a Line

To parametrize a line joining two points in the complex plane, we can express it in the form \( z(t) = (1-t)z_0 + tz_1 \), where \( z_0 \) is the starting point, \( z_1 \) is the ending point, and \( t \) is a parameter ranging from 0 to 1.

Step 2: Parametrization for the Line Joining the Origin to Point 1+i

The line joins the origin (0) to the point \(1+i\). Using \( z_0 = 0 \) and \( z_1 = 1+i \), the parametrization is: \[ z(t) = (1-t) imes 0 + t(1+i) = t(1+i) \].

Step 3: Parametrization for the Line Joining Point 1 to Point 1+i

The line joins point 1 to \(1+i\). Using \( z_0 = 1 \) and \( z_1 = 1+i \), the parametrization is: \[ z(t) = (1-t) imes 1 + t(1+i) = 1 + ti \].

Step 4: Parametrization for the Line Joining Point i to Point 1+i

The line joins point \(i\) to \(1+i\). Using \( z_0 = i \) and \( z_1 = 1+i \), the parametrization is: \[ z(t) = (1-t) imes i + t(1+i) = i + t - ti \].

Step 5: Parametrization for the Line Joining Point 2 to Point 1+i

The line joins point 2 to \(1+i\). Using \( z_0 = 2 \) and \( z_1 = 1+i \), the parametrization is: \[ z(t) = (1-t) imes 2 + t(1+i) = 2 - 2t + t + ti = 2 - t + ti \].

Key Concepts

Parametrization

Parametrization is a powerful concept that allows us to express a geometric object, like a line, using a parameter. Often, the parameter is denoted by a variable such as \( t \). This makes it easier to handle in problems involving motion and time, as well as providing a smooth transition from one point to another. In complex analysis, parametrize means to express points along a line in the complex plane with an equation. Here, the parametric equation of a line joining two points \( z_0 \) and \( z_1 \) in the complex plane is given by:\[z(t) = (1-t)z_0 + tz_1\]

• \( z_0 \) is the starting point.
• \( z_1 \) is the end point.
• \( t \) is a parameter ranging from 0 to 1.

The value of \( t \) changes smoothly. When \( t = 0 \), \( z(t) \) is the starting point \( z_0 \). When \( t = 1 \), \( z(t) \) becomes the end point \( z_1 \). This equation ensures a linear change and is applicable in many scenarios.

Complex Plane

The complex plane, sometimes referred to as the Argand plane, is a fundamental idea in complex analysis. It provides an intuitive way to visualize complex numbers. Imagine it as a two-dimensional plane similar to the Cartesian coordinate system. In this setup, each complex number \( z = x + yi \) is represented by a point on the plane:

• The horizontal axis (real axis) represents the real part \( x \) of the complex number.
• The vertical axis (imaginary axis) represents the imaginary part \( y \).

Each point or position on this plane corresponds uniquely to a complex number. This visualization helps in performing operations on complex numbers and understanding their geometric interpretation, making concepts like addition, subtraction, and even multiplication more accessible. For example, adding two complex numbers can be seen as vector addition on this plane.

Lines in Complex Numbers

Lines in the complex plane are similar to lines in geometry, but with complex numbers as coordinates. When defining a line between two points, say \( z_0 \) and \( z_1 \), it can be visualized as a straight path on the plane. The equation for this would be:\[z(t) = (1-t)z_0 + tz_1\]Here, \( t \) acts as an interpolating factor running from 0 to 1, smoothly transitioning the point along the line.

• When \( t = 0 \), the point is at \( z_0 \).
• When \( t = 1 \), the point reaches \( z_1 \).
• For \( 0 < t < 1 \), the point \( z(t) \) lies somewhere between \( z_0 \) and \( z_1 \).

This concept makes it easier to map movements or interpolate positions between two complex points, improving the understanding and manipulation of complex numbers in geometric terms.