Functional calculus

Functional calculus is a mathematical framework that allows us to apply functions—like square roots, exponentials, or more complex ones—to operators, such as matrices or linear transformations, in a consistent way. It extends the idea of plugging numbers into functions to more abstract objects, enabling us to analyze and work with their properties systematically. This approach is essential in areas like quantum mechanics and differential equations, where understanding how functions operate on operators helps solve complex problems and reveal deeper insights into their behavior.