What is Rotation in Complex Analysis?

In complex analysis, rotation is fundamentally a convenient method used to relate complex numbers and the angles they make. This method is widely applied because it provides a powerful geometric interpretation for operations involving complex numbers.

Understanding Complex Number Rotation

At its core, rotation in the complex plane allows us to represent the geometric action of turning a point (represented by a complex number) around the origin.

• Relating Numbers and Angles: As highlighted by the reference, rotation is a key technique to connect the algebraic representation of complex numbers (like z = x + iy) with their geometric properties, particularly their argument (the angle they make with the positive real axis).
• No New Concepts: The reference notes that this method involves no new concepts, meaning it leverages existing mathematical ideas about multiplication and angles, simply applying them within the structure of complex numbers.

How Rotation Works

The mechanism for performing a rotation in the complex plane involves multiplication.

When you multiply a complex number z by a complex number of the form e^(iθ) (or cos(θ) + i sin(θ)), the result is a new complex number that corresponds to rotating the point representing z counterclockwise around the origin by the angle θ.

Let's break this down:

1. A Complex Number as a Vector: A complex number z = x + iy can be viewed as a vector from the origin (0,0) to the point (x,y) in the complex plane.
2. Multiplication by a Unit Complex Number: A complex number on the unit circle (like e^(iθ)), has a magnitude (or modulus) of 1. Multiplying z by such a number scales z by 1 (no change in length) and rotates z by the argument of the multiplier (θ).
3. The Result: The product z * e^(iθ) gives a new complex number representing the point z rotated by angle θ.

Operation	Effect	Complex Form
Multiplication by e^(iθ)	Rotates a complex number counterclockwise	z' = z * e^(iθ)

This elegant connection between complex multiplication and geometric rotation makes complex numbers invaluable for solving problems in geometry, physics, and engineering involving rotations.

Practical Insights

• Clockwise Rotation: To rotate clockwise by an angle θ, you multiply by e^(-iθ) (or cos(θ) - i sin(θ)).
• Rotation about a Point Other Than the Origin: To rotate a complex number z about a point c by angle θ, you first translate z so c is the origin (z - c), then rotate the result ((z - c) * e^(iθ)), and finally translate it back by c ((z - c) * e^(iθ) + c).

This shows how rotation in complex analysis provides a straightforward algebraic method to perform geometric transformations, relating complex numbers and the angles involved in these transformations.