analytic continuation
Noun
Meaning
(mathematical analysis) The practice of extending analytic functions.
(mathematical analysis) An extension of an analytic function which is itself analytic.
Source: en.wiktionary.orgExamples
Hasse's conjecture affirms that the L-function admits an analytic continuation to the whole complex plane and satisfies a functional equation relating, for any s, L(E, s) to L(E, 2 − s).
Instead of the above ad hoc analytic continuation assumption, non-convergent sums and integrals are computed using Euler–Maclaurin summation with a regularizing function (e.
The formula also holds for all α ≠ −n, −n − 1, … by analytic continuation, but then the function and its Fourier transforms need to be understood as suitably regularized tempered distributions.
The functional equation was established by Riemann in his 1859 paper On the Number of Primes Less Than a Given Magnitude and used to construct the analytic continuation in the first place.