A Mobius transformation is a fundamental concept in complex analysis. It's defined as f of z equals a z plus b divided by c z plus d, where a, b, c, and d are complex numbers, with the constraint that a d minus b c is not equal to zero. This transformation maps the extended complex plane to itself and has remarkable properties, including mapping circles and lines to circles and lines.
Mobius transformations have several remarkable properties. First, they are bijective mappings, meaning they are one-to-one and onto. Second, they are conformal, which means they preserve angles between intersecting curves. Third, they map circles to circles, and lines to either circles or lines. The domain and codomain is the extended complex plane, which includes the complex numbers plus the point at infinity.
Any Mobius transformation can be decomposed into three basic operations. First is translation, which shifts points by adding a constant b. Second is scaling and rotation, which multiplies by a complex number a. Third is inversion, which maps z to one over z. This inversion is the most distinctive operation, as it maps the interior of the unit circle to the exterior and vice versa.
Let's look at a specific example: the inversion transformation f of z equals one over z. This maps one to one, i to negative i, two to one half, and infinity to zero. Notice that the unit circle maps to itself under this transformation. Points inside the unit circle map to points outside, and vice versa. This demonstrates the circle-preserving property of Mobius transformations.
To summarize what we've learned about Mobius transformations: They are functions that map the extended complex plane to itself while preserving angles and mapping circles to circles. Any Mobius transformation can be decomposed into basic operations of translation, scaling, and inversion. These transformations form a fundamental tool in complex analysis and have wide applications in geometry and mathematical physics.