Problem 2 - Complex Analysis for Mathematics and Engineering

Question

Sketch the curve \(z(t)=t^{2}+2 t+i(t+1)\) (a) for \(-1 \leq t \leq 0\). (b) for \(1 \leq t \leq 2\).

Step-by-Step Solution

Answer

Plot the points and connect them to form two segments from \((-1, 0)\) to \((0, 1)\) and from \((3, 2)\) to \((8, 3)\).

Step 1: Express Complex Function

The given function is \( z(t) = t^2 + 2t + i(t + 1) \). This can also be expressed as \( z(t) = (t^2 + 2t) + i(t + 1) \), where the real part is \( t^2 + 2t \) and the imaginary part is \( t + 1 \).

Step 2: Evaluate for Interval \(-1 \leq t \leq 0\)

To sketch the curve for part (a), substitute values of \( t \) from \(-1\) to \(0\) into both the real and imaginary parts. Calculate the following points:- At \( t=-1 \), real part: \((-1)^2 + 2(-1) = -1\), imaginary part: \(-1 + 1 = 0\). Hence, \( z(-1) = -1 \).- At \( t=0 \), real part: \(0^2 + 2(0) = 0\), imaginary part: \(0 + 1 = 1\). Hence, \( z(0) = i \).

Step 3: Plot Points for \(-1 \leq t \leq 0\)

In the complex plane, plot the points derived from the evaluations:- \( (-1, 0) \) for \( t = -1 \)- \( (0, 1) \) for \( t = 0 \)Draw a curve connecting these points to represent the graph between \( t = -1 \) and \( t = 0 \).

Step 4: Evaluate for Interval \(1 \leq t \leq 2\)

To sketch the curve for part (b), substitute values from \(1\) to \(2\):- At \( t=1 \), real part: \(1^2 + 2(1) = 3\), imaginary part: \(1 + 1 = 2\). Hence, \( z(1) = 3 + 2i \).- At \( t=2 \), real part: \(2^2 + 2(2) = 8\), imaginary part: \(2 + 1 = 3\). Hence, \( z(2) = 8 + 3i \).

Step 5: Plot Points for \(1 \leq t \leq 2\)

In the complex plane, plot the points:- \( (3, 2) \) for \( t = 1 \)- \( (8, 3) \) for \( t = 2 \)Draw a curve connecting these points to represent the part of the graph between \( t = 1 \) and \( t = 2 \).

Key Concepts

Complex Plane

The complex plane is a way to envision complex numbers by assigning a point on a two-dimensional graph. In this graph, the x-axis represents the real part of the complex number, while the y-axis represents the imaginary part.
This offers a visual perspective where complex numbers are treated like coordinates.

• The horizontal axis (real axis) is where you plot the real component of the complex number, such as the term \(t^2 + 2t\) in the exercise.
• The vertical axis (imaginary axis) is for the imaginary component, like \(i(t+1)\).

Knowing how to plot these points allows you to visualize the behavior of complex functions more easily. When you join these plotted points with a line or curve, it translates algebraic expressions into geometric representations on this complex plane.

Complex Numbers

Complex numbers consist of a real part and an imaginary part. The standard format is \(a + bi\), where \(a\) represents the real component, and \(b\) is the coefficient in front of the imaginary unit \(i\).
The imaginary unit is defined as \(i = \sqrt{-1}\), which is fundamental to complex arithmetic.

• The real part, \(a\), can be any number and defines the position on the real axis.
• The imaginary part, \(b\), shifts the number along the imaginary axis.

In the exercise, we express the function \(z(t) = (t^2 + 2t) + i(t + 1)\), where each complex number has distinct real and imaginary components based on \(t\). By evaluating this at specific intervals, like \(-1 \leq t \leq 0\) and \(1 \leq t \leq 2\), you can see how both parts vary, influencing the shape on the complex plane.

Graphing

Graphing complex functions involves plotting complex numbers on the complex plane at various values of the variable, here denoted as \(t\). This often entails:

• Evaluating the function at specific points of \(t\) to determine exact coordinates in the complex plane.
• Noting changes in both the real and imaginary parts as \(t\) varies to sketch a curve reflecting the function's behavior.

Consider the exercise's function \(z(t) = t^2 + 2t + i(t+1)\). For \(-1 \leq t \leq 0\), you plot points like \((-1, 0)\), then continue for \(1 \leq t \leq 2\) with points like \(3 + 2i\).
The step-by-step calculation allows one to methodically chart the function by first separating it into its real and imaginary parts before mapping them. Graphing helps you understand complex dynamics in a visual format, crucial for observing patterns and predicting future values.