Problem 15 - Advanced Engineering Mathematics

Question

The mapping \(w=z+4 i\) is a translation which maps the circle \(|z|=1\) to a circle of radius \(r=1\) and with center \(w=4 i .\) This circle may be described by \(|w-4 i|=1.\)

Step-by-Step Solution

Answer

The mapping shifts the circle \(|z| = 1\) to \(|w - 4i| = 1\).

Step 1: Understanding the Given Mapping

The mapping given is \(w = z + 4i\). This represents a translation in the complex plane by 4 units in the imaginary direction.

Step 2: Identifying the Original Circle

The original circle is represented by \(|z| = 1\). This means the circle has a center at the origin \((0,0)\) and a radius of 1.

Step 3: Performing the Translation

The translation \(w = z + 4i\) moves every point of the circle \(|z| = 1\) directly upward in the imaginary axis by 4 units. This means the whole circle is shifted without changing its size or shape.

Step 4: Describing the New Circle

After the translation, the circle retains its radius of 1, but its center moves to \(w = 4i\). In mathematical terms, this translated circle is described by the equation \(|w - 4i| = 1\).

Step 5: Verification and Interpretation

The equation \(|w - 4i| = 1\) confirms that the center is at \(4i\) and the radius is 1, which matches the description of the translated circle after the mapping.

Key Concepts

Complex Plane

The complex plane is a two-dimensional plane where each point represents a complex number. Imagine a grid where the horizontal axis is the real axis and the vertical axis is the imaginary axis. Complex numbers are written in the form \( z = x + yi \), where \( x \) represents the real part and \( y \) represents the imaginary part.

In the complex plane:

• The point \( (x, 0) \) on the real axis represents a real number.
• The point \( (0, yi) \) on the imaginary axis represents a "purely" imaginary number.
• Any other point \( (x, yi) \) represents a complex number \( x + yi \).

This visualization helps us see how complex numbers can be manipulated geometrically, such as rotations, reflections, and translations.

Complex Mapping

Complex mapping involves transforming points in the complex plane using mathematical operations. It can take simple forms like addition or multiplication, or more complex forms like exponentiation or logarithms.

For example, consider the mapping \( w = z + 4i \), which adds a constant imaginary part to \( z \). This results in a translation of points in the complex plane by 4 units upward. Each point \( z \) is transformed into a new point \( w \) following this rule.

Such mappings are useful in understanding conformal mappings, where the angle between curves is preserved, and in solving differential equations through geometric interpretations.

Translation in Complex Numbers

A translation in complex numbers involves moving every point in the plane by the same vector. In complex terms, a translation is achieved by adding a constant complex number to all points.

• For example, the mapping \( w = z + 4i \) translates each point \( z \) by 4 units in the imaginary direction, resulting in \( w \).
• This operation does not change the size or shape of geometric figures like circles or rectangles; it merely shifts their position.
• The circle \( |z| = 1 \), centered at the origin, becomes \( |w - 4i| = 1 \) after translation, indicating a move to a new center at \( 4i \).

Translations simplify dynamics in the complex plane by allowing us to reposition objects without altering their intrinsic properties, such as radius or symmetry.