Generalized Functions

Generalized functions, also known as distributions, are mathematical tools that extend the concept of regular functions to include more singular objects, like impulses or point sources. They allow us to rigorously analyze situations where classical functions fail, such as modeling a concentrated force or an instantaneous impulse. Think of them as a way to handle "infinite" or "spike-like" behavior mathematically, enabling precise analysis in fields like physics and engineering. Instead of being ordinary functions, they are defined in terms of how they interact with smooth test functions, making complex problems more manageable.