Problem 14 - Advanced Engineering Mathematics

Question

Show that if a curve in the \(w\) -plane is parametrized by \(w=w(t)\). \(a \leq t \leq b\), and arg \(w^{\prime}(t)\) is constant, then the curve is a line seg ment. [Hint: If \(w(t)=u(t)+i v(t)\), then \(\tan \left(\arg w^{\prime}(t)\right)=d v^{\prime} d u\).

Step-by-Step Solution

Answer

The curve \( w(t) \) is a line segment because it satisfies the line equation \( v = cu + d \) with constant \( c \) and \( d \).

Step 1: Introduce the Parametrization

Consider the parametrization of the curve in the complex plane, given by \( w(t) = u(t) + iv(t) \). We need to show that if the argument of its derivative is constant, then \( w(t) \) traces out a line segment.

Step 2: Differentiate the Parametric Equation

Differentiate \( w(t) \) to obtain \( w'(t) = u'(t) + iv'(t) \). This derivative represents the velocity vector of the curve in the complex plane.

Step 3: Analyze the Constant Argument

The hint provides that \( \tan(\arg(w'(t))) = \frac{v'(t)}{u'(t)} \). If \( \arg(w'(t)) \) is constant, then \( \frac{v'(t)}{u'(t)} = c \) for some constant \( c \).

Step 4: Solve for \( v'(t) \)

Rearranging the equation from Step 3, we have \( v'(t) = c \cdot u'(t) \). This suggests that the derivatives of \( u(t) \) and \( v(t) \) are proportional.

Step 5: Integrate the Derivative Relations

Integrate both sides to find the functional forms of \( u(t) \) and \( v(t) \). From \( v'(t) = c \cdot u'(t) \), integrating gives \( v(t) = c u(t) + d \), where \( d \) is a constant of integration.

Step 6: Conclude with a Line Segment Equation

The resulting equation \( v(t) = c u(t) + d \) is the equation of a straight line in the \( (u, v) \)-plane. Therefore, as \( t \) runs from \( a \) to \( b \), the curve described by \( w(t) \) traces out a line segment.

Key Concepts

Curve Parametrization

When we talk about curve parametrization in complex analysis, we're essentially discussing a method to describe a curve using a single parameter, usually denoted as \( t \). Here, the curve is given as \( w(t) = u(t) + iv(t) \), where \( u(t) \) and \( v(t) \) are real-valued functions representing the real and imaginary parts of \( w(t) \), respectively.


This parametrization allows us to track every point along the curve as \( t \) varies. Think of \( t \) as a time variable, where at each moment \( t \), the curve has a specific point \( w(t) \) in the complex plane. This concept is powerful because instead of dealing with the curve as an abstract set of points, we have a concrete formula to work with, making analysis and calculations much simpler.

• By differentiating \( w(t) \), we can find its rate of change which is crucial for understanding the curve's properties.
• Parametrization also facilitates the application of calculus, particularly when determining the curve's length, tangent vectors, and more.

Constant Argument

A key part of our problem involves the concept of 'constant argument'. In the context of a complex function or curve, the argument of a complex number \( w(t) = u(t) + iv(t) \) is the angle \( \theta \) formed by the vector \( (u(t), v(t)) \) with the positive real axis. This is given by \( \tan(\theta) = \frac{v'(t)}{u'(t)} \).

If the argument of \( w'(t) \) is constant, it means that this angle does not change as \( t \) changes. As a result, the line \( w(t) \) follows a straight path in the complex plane. That's similar to painting a line with a constant brush angle, where the direction does not vary.

• Understanding constant argument helps us identify straight lines and symmetry in complex functions.
• In the exercise solution, knowing the argument of the derivative is constant directly leads to realizing that the proportional relationship between \( u'(t) \) and \( v'(t) \) persists.

Line Segment

The final part of our exercise concludes with showing that a curve with a constant argument results in a line segment. This is the geometric path traced by the parametric equation \( w(t) = u(t) + iv(t) \) from \( t = a \) to \( t = b \). When \( v(t) = cu(t) + d \) arises from integrating \( v'(t) = cu'(t) \), we recognize this as the equation of a line in the plane.


Each \( t \) corresponds to a point on the straight line, hence the term "line segment." It’s important to note that in the complex plane, this straight line is represented in terms of \( w(t) \), showing a clear and intuitive relationship between \( u(t) \) and \( v(t) \).

• Recognizing line segments is essential for geometric intuition in complex analysis as it relates simple algebraic expressions with visual representation.
• Even though the path might be simple – a line segment – deriving this from a complex constraint emphasizes how powerful the tools of complex analysis can be.