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hide. 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 ]
Radius of convergence. In mathematics, the radius of convergence of a power series is the radius of the largest disk at the center of the series in which the series converges. It is either a non-negative real number or . When it is positive, the power series converges absolutely and uniformly on compact sets inside the open disk of radius equal ...
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. See for example, the binomial series.
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).
Mathematics of three-phase electric power. One voltage cycle of a three-phase system, labeled 0 to 360° (2π radians) along the time axis. The plotted line represents the variation of instantaneous voltage (or current) with respect to time. This cycle repeats with a frequency that depends on the power system.
A power series is here defined to be an infinite series of the form where jj1jn is a vector of natural numbers, the coefficients a(j1, …, jn) are usually real or complex numbers, and the center cc1cn and argument xx1xn are usually real or complex vectors. The symbol is the product symbol, denoting multiplication.
Notice that Abel's test implies in particular that the radius of convergence is at least 1. It can also be applied to a power series with radius of convergence R ≠ 1 by a simple change of variables ζ = z/R. [2] Notice that Abel's test is a generalization of the Leibniz Criterion by taking z = −1. Proof of Abel's test: Suppose that z is a ...
exists there are three possibilities: if L > 1 the series converges (this includes the case L = ∞) if L < 1 the series diverges. and if L = 1 the test is inconclusive. An alternative formulation of this test is as follows. Let { an } be a series of real numbers. Then if b > 1 and K (a natural number) exist such that.