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For a power series f defined as: = = (),where a is a complex constant, the center of the disk of convergence,; c n is the n-th complex coefficient, and; z is a complex variable.; The radius of convergence r is a nonnegative real number or such that the series converges if
In mathematics, the Cauchy–Hadamard theorem is a result in complex analysis named after the French mathematicians Augustin Louis Cauchy and Jacques Hadamard, describing the radius of convergence of a power series. It was published in 1821 by Cauchy, [1] but remained relatively unknown until Hadamard rediscovered it. [2]
The sum of a power series with a positive radius of convergence is an analytic function at every point in the interior of the disc of convergence. However, different behavior can occur at points on the boundary of that disc. For example:
Note that sometimes a series like this is called a power series "around p", because the radius of convergence is the radius R of the largest interval or disc centred at p such that the series will converge for all points z strictly in the interior (convergence on the boundary of the interval or disc generally has to be checked separately).
3.3.3 Radius of convergence of power series. 3.3.4 Laurent series expansion. ... is the interior of the set of points of absolute convergence of some power series in ...
Excluding these cases, the ratio test can be applied to determine the radius of convergence. If p < q + 1 then the ratio of coefficients tends to zero. This implies that the series converges for any finite value of z and thus defines an entire function of z. An example is the power series for the exponential function.
The utility of Abel's theorem is that it allows us to find the limit of a power series as its argument (that is, ) approaches from below, even in cases where the radius of convergence, , of the power series is equal to and we cannot be sure whether the limit should be finite or not.
That theorem has its main interest in the case that the power series has radius of convergence exactly 1: if the radius of convergence is greater than one, the convergence of the power series is uniform for r in [0,1] so that the sum is automatically continuous and it follows directly that the limit as r tends up to 1 is simply the sum of the a n.