Analytic continuation
Analysis
The extension of a complex function beyond the region where its original formula converges, to a larger domain on which it remains complex-differentiable.
What is usually left out
The extension, where it exists, is unique — which is what makes it legitimate rather than arbitrary. This matters for the zeta function: ∑ 1/nˢ diverges for s = −1, yet ζ(−1) = −1/12 is a well-defined value of the continued function. That value is not the sum of the divergent series, and treating it as one is a common misreading.