Aug 112013

这里指的是复分析中关于幂级数的 Abel’s theorem, 目的是讨论幂级数在收敛圆周的性态.

\( D=\{z\in\Bbb C:|z|<1\} \). Let \(f(z)=\sum\limits_{n=0}^\infty a_n z^n(a_n,z\in\Bbb C)\)be a power series, and the radius of convergence of \(f(z)\) is \(1\), \(\sum\limits_{n=0}^\infty a_n =s\). we cannot conclude that

\[\lim_{D\ni z\to1 }f(z)= s.\]


In 1916, Sierpiński constructed a power series with radius of convergence equal to \(1\), also converging on every point of the unit circle, but with the property that \(f\) is unbounded near \(z=1\).

Sierpiński 的例子很复杂, 在一本法文书上可以找到.

For odd \(n\) let  \(p_n = 1\cdot 3\cdot 5\cdots n\), For even \(n\) set \(p_n=2p_{n-1}\). Define

\[ f(z)=\sum_{n=1}^{\infty}\frac{(-1)^n}nz^{p_n}. \]