Le theoreme de Picard-Borel et la theorie des fonctions meromorphes
Le theoreme de Picard-Borel et la theorie des fonctions meromorphes
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DOI：
10.2307/3605435
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发表时间：
1930-01
期刊：
影响因子：
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通讯作者：
E. C. Titchmarsh;R. Nevanlinna
中科院分区：
文献类型：
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作者：
E. C. Titchmarsh;R. Nevanlinna
引用的例子(本身很好)自由地散布在书中，作为读者的练习。也有一种倾向，认为似乎不值得这些描述的“显而易见的”或“容易证明的”定理而不予考虑。P的底部所陈述的定理的证明。159(卷I)，例如，会受到评审员的欢迎。又一次，关于一个级数的收敛圆在其周长上至少有一个由级数定义的函数的奇点的定理，不应该在两行中被认为是“明显的”(第458页)。决不能从我们选择了一两个要点进行批评就推断我们对整本书的态度是不利的。情况远非如此。这本书的真正优点对以前版本的读者来说已经足够清楚了，这里没有必要详细说明它们。然而，我们可以提到一两点。用“带余数的泰勒定理”来处理零点和极点是值得推荐的，因为它不涉及不必要的限制过程，其效果随后要通过呼吁幂函数级数的性质来抵消。由极限(特别是定积分)定义的函数的正则性定理的证明通过使用“柯西定理的逆”变得特别简单。一般理论是自由地通过有教育意义的例子来说明的，一些被纳入文本，一些被设置为读者的练习。最后，这本书不仅称赞自己是解析函数的一般性质的系统描述，而且也是关于各种特殊主题的有价值的信息来源，在其他地方很难找到令人满意的描述。人工智能。
references to the examples which (excellent in themselves) are freely scattered through the book as exercises for the reader. There is also a tendency to dismiss as " obvious " or " easy to prove " theorems which do not seem to merit these descriptions. A proof of the theorem stated at the bottom of p . 159 (Vol. I), for example, would have been welcomed by the reviewer. Again the theorem tha t the circle of convergence of a power series has on its circumference a t least one singularity of the function defined by the series should scarcely have been dismissed in two lines as " obvious " (p. 458). I t must not be inferred from the fact that we have chosen one or two points for criticism tha t our att i tude to the book as a whole is unfavourable. This is far from being the case. The very real merits of the book are sufficiently well known to readers of previous editions, and there is no need to give a detailed account of them here. We may, however, mention one or two points. The treatment of zeros and poles by means of " Taylor's theorem with remainder " is to be commended, since it does not involve an unnecessary limiting process whose effect has subsequently to be cancelled by an appeal to the properties of power series. The proofs of the theorems on regularity of functions defined by limits (in particular by definite integrals) are rendered particularly simple by the use of the " converse of Cauchy's theorem." The general theory is freely illustrated by instructive examples, some incorporated in the text and some set as exercises for the reader. Finally, the book commends itself not only as a systematic account of the general properties of analytic functions, but also as a valuable source of information on a variety of special topics of which a satisfactory account is not easy to find elsewhere. A. E. I .