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Let an analytic element be a power series associated with an open disc of the complex plane over which the series is convergent. W.l.o.g. assume the series is a Taylor expansion about the center of the disc. It is easy to show that analytic continuation is an equivalence relationship between analytic elements.

Given two analytic elements (𝑈,𝑓) and (𝑉,𝑔), how can we tell whether they are in the same class? Obviously if f and g are both polynomials of degrees at most n we could see if the two are in the same class by evaluating at n+1 points. On the other extreme: even comparing coefficients of two general series may take infinite steps. So let me bifurcate the question:

• Without concern for computability, what can we say in general on how to decide if two analytic elements are in the same class?

• What restrictions do we need to restore computability to this question?

Let an analytic element be a power series associated with an open disc of the complex plane over which the series is convergent. W.l.o.g. assume the series is a Taylor expansion about the center of the disc. It is easy to show that analytic continuation is an equivalence relationship between analytic elements.

Given two analytic elements $(U, f)$ and $(V, g)$, how can we tell whether they are in the same class? Obviously if $f$ and $g$ are both polynomials of degrees at most $n$ we could see if the two are in the same class by evaluating at $n + 1$ points. On the other extreme: even comparing coefficients of two general series may take infinite steps. So let me bifurcate the question:

• Without concern for computability, what can we say in general on how to decide if two analytic elements are in the same class?

• What restrictions do we need to restore computability to this question?

Let an analytic element be a power series associated with an open disc of the complex plane over which the series is convergent. W.l.o.g. assume the series is a Taylor expansion about the center of the disc. It is easy to show that analytic continuation is an equivalence relationship between analytic elements. Is there a decision procedure to determine if an analytic element is in a particular equivalence class? Put equivalently, given two analytic elements, is there a way to determine if they are analytic continuations of each other?

Let an analytic element be a power series associated with an open disc of the complex plane over which the series is convergent. W.l.o.g. assume the series is a Taylor expansion about the center of the disc. It is easy to show that analytic continuation is an equivalence relationship between analytic elements.

Given two analytic elements (𝑈,𝑓) and (𝑉,𝑔), how can we tell whether they are in the same class? Obviously if f and g are both polynomials of degrees at most n we could see if the two are in the same class by evaluating at n+1 points. On the other extreme: even comparing coefficients of two general series may take infinite steps. So let me bifurcate the question:

• Without concern for computability, what can we say in general on how to decide if two analytic elements are in the same class?

• What restrictions do we need to restore computability to this question?

Let an analytic element be a power series associated with a open disc of the complex plane over which the series is convergent. W.l.o.g. assume the series is a Taylor expansion about the center of the disc. It is easy to show that analytic continuation is an equivalence relationship between analytic elements. Is there a decision procedure to determine if an analytic element is in a particular equivalence class? Put equivalently, given two analytic elements, is there a way to determine if they are analytic continuations of each other?

Let an analytic element be a power series associated with an open disc of the complex plane over which the series is convergent. W.l.o.g. assume the series is a Taylor expansion about the center of the disc. It is easy to show that analytic continuation is an equivalence relationship between analytic elements. Is there a decision procedure to determine if an analytic element is in a particular equivalence class? Put equivalently, given two analytic elements, is there a way to determine if they are analytic continuations of each other?