Analytic continuation is ill-posed, but becomes merely ill-conditioned (although with an infinite condition number) if it is known that the function in question is bounded in a given region of the complex plane. In an annulus, the Hadamard three-circles theorem implies that the ill-conditioning is not too severe, and we show how this explains the effectiveness of Chebfun and related numerical methods in evaluating analytic functions off the interval of definition. By contrast, we show that analytic continuation is far more ill-conditioned in a strip or a channel, with exponential loss of digits of accuracy at the rate $$\exp (-\pi x/2)$$ exp ( - π x / 2 ) as one moves along. The classical Weierstrass chain-of-disks method loses digits at the faster rate $$\exp (-ex)$$ exp ( - e x ) .

中文翻译：

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量化分析延续的病态

解析延拓是病态的，但如果已知所讨论的函数在复平面的给定区域中是有界的，则它只是病态的（尽管具有无限的条件数）。在环中，Hadamard 三圆定理暗示病态条件不太严重，我们展示了这如何解释 Chebfun 和相关数值方法在评估定义区间外的解析函数方面的有效性。相比之下，我们表明解析延拓在条带或通道中的病态要严重得多，精度数字的指数损失以 $$\exp (-\pi x/2)$$ exp ( - π x / 2 ) 随着一个人的前进。经典的 Weierstrass 磁盘链方法以更快的速度丢失数字 $$\exp (-ex)$$ exp ( - ex ) 。