# What is the limit of...

$$\prod\limits_{n\ge2}\left(1+\frac{1}{n^2}\right)$$

If a complex function is analytic at all finite points of the complex plane C, then it is said to be entire, sometimes also called "integral" (Knopp 1996, p. 112).

Wolfram MathWorld - Entire Function

A complex function is said to be analytic on a region $\mathbb R$ if it is complex differentiable at every point in $\mathbb R$

Wolfram MathWorld - Analytic Function

Note, 在区域上某个点可微，即不管以何种方式／路径趋近，极限都要存在。这一点和二元函数在某个点极限存在的定义十分类似。

《高等数学》（下册）P54

$f\;'(0) = \lim_{z\to0}\frac{f(z)-f(0)}{z-0} = \lim_{z\to0}\frac{f(z)}{z} \qquad...\qquad (1)$


$$\lim_{r\to0} \frac{f(re^{i\theta})}{re^{i\theta}}$$

Proof

$$\sum_{n \in \mathbb Z^*} \frac{1}{n^{h+1}}$$
（这也是整函数$\sin(\pi z)$的亏格(genus)，应用Weierstrass factorization定理需要它）

The Weierstrass factorization theorem asserts that entire functions can be represented by a product involving their zeroes. In addition, every sequence tending to infinity has an associated entire function with zeroes at precisely the points of that sequence.

Wikipedia - Weierstrass factorization theorem

$$\sin(\pi z)=z e^{g(z)} \prod_{n \in \mathbb Z^*} \left(1-\frac{z}{n} \right) e^{z/n}$$
$g(z)$是一个待定的整函数，取对数导数[1]（这样我们就能将积写为和）

$\pi \cot (\pi z)=\frac{1}{z}+g'(z)+ \sum_{n \in \mathbb Z^*} \left(\frac{1}{z-n}+\frac{1}{n}\right)\qquad...\qquad(2)$


$2\pi \cot (2\pi z) = \pi \frac{\cos^2(\pi z) - \sin^2(\pi z)}{\sin (\pi z)\cos (\pi z)} = \pi \cot (\pi z) + \pi\cot \left(\pi \left(z+\tfrac{1}{2}\right)\right)$


$s(z) = \frac{1}{z} + \sum_{n\neq 0} \left(\frac{1}{z-n} + \frac{1}{n}\right) = \lim_{N\to\infty} \sum_{n=-N}^N \frac{1}{z-n}$


\begin{align}s(z) + s\left(z+\tfrac{1}{2}\right) &= \lim_{N\to\infty} \left(\sum_{n=-N}^N \frac{1}{z-n} + \sum_{n=-N}^N\frac{1}{z+\frac{1}{2}-n}\right)\\ &= 2\lim_{N\to\infty} \left(\sum_{n=-N}^N\frac{1}{2z-2n} + \sum_{n=-N}^N \frac{1}{2z+1-2n}\right)\\ &= 2\lim_{N\to\infty} \sum_{k=-(2N+1)}^{2N} \frac{1}{2z-k}\\ &= 2s(2z)\end{align}


$$g(z) + g\left(z+\tfrac{1}{2}\right) = 2g(2z)$$

$2 f(x_0) = 2f(x_k) = 2f(2x_{k+1}) = f(x_{k+1}) + f\left(x_{k+1}+\tfrac{1}{2}\right)$


\begin{align}\sin(\pi z) &= \pi z \prod_{n \in \mathbb Z^*} \left( 1-\frac{z}{n} \right) e^{z/n} \\&=\pi z \prod_{n=1}^\infty \left(1-\frac{z^2}{n^2}\right)\end{align}

$\sin(z)=z \prod_{n=1}^\infty \left(1-\frac{z^2}{n^2 \pi^2} \right)\qquad...\qquad(3)$

\begin{align}\sinh(z)=-i \sin(iz) &= -i \cdot i z \prod_{n=1}^\infty \left(1+\frac{z^2}{\pi^2 n^2} \right) \\&= z \prod_{n=1}^\infty \left(1+\frac{z^2}{\pi^2 n^2} \right)\end{align}

$\frac{\sinh(\pi x)}{\pi x}=\prod_{n=1}^\infty \left(1+\frac{x^2}{n^2} \right)\qquad...\qquad(4)$

$$\prod_{n=1}^\infty \left(1+\frac{1}{n^2} \right)=\frac{\sinh(\pi)}{\pi}$$

\begin{align}\prod_{n=2}^\infty \left(1+\frac{1}{n^2} \right)&=\frac{\prod_{n=1}^\infty \left(1+\frac{1}{n^2} \right)}{1+\tfrac{1}{1}}\\&=\frac{\sinh \pi}{2\pi}\end{align}

[1] Wikipedia - Logarithmic derivative
[2] Wolfram MathWorld - Entire Function
[3] Wolfram MathWorld - Analytic Function
[4] Wikipedia - Weierstrass factorization theorem

声明: 本文为0xBBC原创, 转载注明出处喵～