It is often stated that a single-valued analytic function f(z) is uniquely and
completely determined if (1) it is analytic at all points of a convergent sequence of
points in the complex plane and at their limit point and (2) one is given the points
of the sequence and the values of f(z) at each of these points.


Let z(1),z(2),...,z(n)... be a convergent sequence of complex numbers which are
strictly decreasing in absolute value as n increases, and whose limit point is zero.
Let f(z) be analytic at all the points of this sequence and at their limit point.
Supposing that for each positive integer i one is given z(i) and f(z(i)). Does there
then always exist a unique power series P(z) centered at zero such that (3) the radius
of convergence R of P(z) is positive (or infinite) and (4) if k is any positive integer
for which the absolute value of z(k) is less than R, P(z(k))=f(z(k))?


If such a unique power series exists, how do we obtain its coefficients from the
data we are given? One can set up an equation for these unknown coefficients involving
two infinite column vectors and an infinite Vandermonde matrix. The rows of the matrix
are all of the form 1,z(j),z(j)^2,z(j)^3...where j is a positive integer. But I do not
know what conditions are needed to insure that such matrices have a unique inverse.



A question that arises in trying to make mathematically precise a well known informal statement about analytic functions
If I am not misunderstanding your question, you read off the power series pretty much directly from the given data. You know f(0). You also know f'(0) by using the definition of the derivative. The higher derivatives can all be determined by using higher order difference equations <http://en.wikipedia.org/wiki/Finite_difference#Higher-order_differences>. Since the function is analytic the taylor series you get this way does converge on some disk...


