Final answer:
Möbius transformations require the condition ad−bc≠0 to ensure they are invertible and do not contain singularities, which is essential for them to function as one-to-one, onto mappings of the extended complex plane.
Step-by-step explanation:
Why do Möbius transformations need ad−bc≠0? This condition is crucial because it ensures that Möbius transformations are well-defined and possess certain properties that make them useful in complex analysis. A Möbius transformation is given by a function f(z) = (az + b) / (cz + d), where a, b, c, and d are complex numbers. The requirement that ad−bc≠0 comes from the need for the transformation to be invertible, meaning there must exist a reverse transformation. If ad−bc = 0, the function would not be invertible, and its denominator could become zero, causing singularities.
To elaborate, a Möbius transformation corresponds to a composition of translations, rotations, dilations (or contractions), and inversions. Each of these operations is invertible on its own; however, when combined into a single function like a Möbius transformation, invertibility is maintained only if the determinant ad−bc is non-zero. This is analogous to invertible matrices in linear algebra where the determinant must also be non-zero. Consequently, this condition helps to avoid singularities and ensures the function maps the extended complex plane to itself in a one-to-one and onto fashion.