Analytic continuation is a mathematical technique used to extend the domain of a given analytic function beyond its original limits. An analytic function is one that can be represented by a power series in a neighborhood of every point in its domain. By finding a new function that agrees with the original function on a certain region, mathematicians can explore values of the function in areas where it was not initially defined. This method is particularly useful in complex analysis, where functions can have singularities or points where they are not defined. A famous example of analytic continuation is the extension of the Riemann zeta function, which is initially defined for complex numbers with a real part greater than 1, to other values through this technique.