1992 Putnam B3

For any pair $(x, y)$ of real numbers, a sequence

$(a_n(x,y))_{n\geq 0}$ is defined as follows:

$$

\begin{aligned}

a_0(x, y) &= x, \\

a_{n+1}(x, y) &= \frac{(a_n(x, y))^2 + y^2}{2}, \qquad \text{for $n \geq 0$}.

\end{aligned}

$$

Find the area of the region

$$

\{ (x, y) | (a_n(x, y))_{n \geq 0}\ \text{converges}\}.

$$

对于任何实数对 $(x, y)$，有一个序列

$(a_n(x,y))_{n\geq 0}$ 定义如下：

$$

\开始{对齐}

a_0(x, y) &= x, \\

a_{n+1}(x, y) &= \frac{(a_n(x, y))^2 + y^2}{2}，\qquad \text{对于 $n \geq 0$}。

\结束{对齐}

$$

求该区域的面积

$$

\{ (x, y) | (a_n(x, y))_{n \geq 0}\ \text{收敛}\}。

$$