analytic continuation

Analytic continuation is a mathematical technique used to extend the domain of a function beyond where it is originally defined. For instance, a function might be well-defined and behave nicely within a certain area, like a circle on a graph. Analytic continuation allows mathematicians to extend that function smoothly to a larger area, finding values and properties even outside the original circle, without introducing contradictions. It's particularly useful in complex analysis, helping to explore functions deeply and revealing connections between seemingly unrelated mathematical concepts.