analytic continuation

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analytic continuation

[‚an·əl′id·ik kən·tin·yü′ā·shən]
(mathematics)
The process of extending an analytic function to a domain larger than the one on which it was originally defined.
References in periodicals archive ?
It is known that [[zeta].sub.K](s) can be analytically continued to C -- {1}, and that at s = 1 it has a simple pole, with residue [[gamma].sub.-1](K), given by the analytic class number formula:
This requires the associated multiplicative function, say f(n), to have special enough properties that the associated Dirichlet series may be analytically continued. In this proposal we continue to develop an approach which requires less of the multiplicative function, linking the original question with the mean value of f.
Notice that the Euler zeta function can be analytically continued to the whole complex plane, and these zeta functions have the values of the Euler numbers at negative integers.
It is a classical and well-known result that [zeta](s), originally defined on the half plane Re(s) > 1, can be analytically continued to a meromorphic function on the entire complex plane with the only pole at s = 1, which is a simple pole with residue 1 [3,6].
This means that [??]([xi]) can be analytically continued along any path which starts from 0 and avoids {[[omega].sub.1] + ...