The idea of analytic continuation

The idea of analytic continuation rests on the notion of analytic function.

An analytic function is a function that is representable by its Taylor series. A Taylor series is a power series, and any power series determines an analytic function inside its domain of convergence.

Since a power series is determined by its coefficients, and since, by Taylor's Theorem, the coefficients are just the higher derivatives evaluated at the center of the power series, it follows that a power series is entirely determined by its values in a very small neighborhood of its center. Likewise, an analytic function is determined by its restriction to a very small neighborhood of any point within its domain of definition.

If two analytic functions agree in some small neighborhood of a point, then it follows that they agree everywhere that both are defined.

Or if two analytic functions agree in some small neighborhood of a point but the domain of one is smaller than the domain of the other, then it is sensible to think of the other as determining an "analytic contination" of the other. For example, the geometric series

1 + z + z^2 + z^3 + z^4 + ...

converges for |z| < 1 and its sum g(z) is an analytic function for |z| < 1.

The geometric series is not convergent or even summable when |z| > 1.

But we know that

g(z) = (1 - z)^{-1}   for   |z| < 1

At first g(z) has a meaning only for |z| < 1; but h(z) = (1 - z)^{-1} has a meaning -- and is analytic -- for all z other than the one value z = 1.

Thus, we may regard h(z) as determining an analytic continuation of g(z). In this way a meaning may be given to the sum of the powers of z. By substituting 2 for z one may obtain the meaning "h(2) = -1" for the sum of the powers of 2. But this latter meaning cannot be separated from the context in which it was obtained. We have not given a meaning to the sum of the powers of 2 apart from thinking of the sum of the powers of 2 as a special case of the sum of the powers of z, performing an analytic continuation (albeit unique), and then specializing.


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