每日彙整: 2017-04-18

樹莓派、樹莓派之學習、樹莓派之教育

時間序列︰生成函數‧漸近展開︰當白努利遇上傅立葉《II》

數學之『進步』怎麼說勒？果是當一個『困難』的問題︰

『巴塞爾問題』是一個著名的『數論問題』，最早由『皮耶特羅‧門戈利』在一六四四年所提出。由於這個問題難倒了以前許多的數學家，因此一七三五年，當『歐拉』一解出這個問題後，他馬上就出名了，當時『歐拉』二十八歲。他把這個問題作了一番推廣，他的想法後來被『黎曼』在一八五九年的論文《論小於給定大數的質數個 數》 On the Number of Primes Less Than a Given Magnitude中所採用，論文中定義了『黎曼ζ函數』，並證明了它的一些基本的性質。那麼為什麼今天稱之為『巴塞爾問題』的呢？因為『此處』這個『巴塞爾』，它正是『歐拉』和『伯努利』之家族的『家鄉』。那麼就這麽樣的一個『級數的和』 \sum \limits_{n=1}^\infty \frac{1}{n^2} = \lim \limits_{n \to +\infty}\left(\frac{1}{1^2} + \frac{1}{2^2} + \cdots + \frac{1}{n^2}\right) 能有什麼『重要性』的嗎？即使僅依據『發散級數』 divergent series 的『可加性』 summable  之『歷史』而言，或又得再過了百年的時間之後，也許早已經是『柯西』之『極限觀』天下後『再議論』的了！！因是我們總該看看『歷史』上『歐拉』自己的『論證』的吧！！

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巴塞爾問題
\sum_{n=1}^{\infty}\frac{1}{n^2} = \frac{\pi^2}{6}

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邏輯之歐拉圖

假使說『三角函數』  \sin{x} 可以表示為 \sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots，那麼『除以』 x 後，將會得到 \frac{\sin(x)}{x} = 1 - \frac{x^2}{3!} + \frac{x^4}{5!} - \frac{x^6}{7!} + \cdots，然而 \sin{x} 的『根』是 x = n\cdot\pi，由於『除以』 x 之緣故，因此 n \neq 0，所以 n = \pm1, \pm2, \pm3, \dots，那麼 \frac{\sin(x)}{x} 應該會『等於』 \left(1 - \frac{x}{\pi}\right)\left(1 + \frac{x}{\pi}\right)\left(1 - \frac{x}{2\pi}\right)\left(1 + \frac{x}{2\pi}\right)\left(1 - \frac{x}{3\pi}\right)\left(1 + \frac{x}{3\pi}\right) \cdots，於是也就『等於』 \left(1 - \frac{x^2}{\pi^2}\right)\left(1 - \frac{x^2}{4\pi^2}\right)\left(1 - \frac{x^2}{9\pi^2}\right) \cdots，若是按造『牛頓恆等式』，考慮 x^2 項的『係數』， 就會有 - \left(\frac{1}{\pi^2} + \frac{1}{4\pi^2} + \frac{1}{9\pi^2} + \cdots \right) = -\frac{1}{\pi^2}\sum_{n=1}^{\infty}\frac{1}{n^2}，然而 \frac{\sin(x)}{x}  之『 x^2 項』的『係數』 是『- \frac{1}{3!} = -\frac{1}{6}』，所以 -\frac{1}{6} = -\frac{1}{\pi^2}\sum \limits_{n=1}^{\infty}\frac{1}{n^2}，於是 \sum \limits_{n=1}^{\infty}\frac{1}{n^2} = \frac{\pi^2}{6}。那麼『歐拉』是『對』的嗎？還是他還是『錯』的呢？？

─── 摘自《【Sonic π】電聲學之電路學《四》之《 V!》‧下中》

 

而今可從 {\tilde{B}}_{2n} (x) = {(-1)}^{n+1} \frac{2 (2n)!}{{(2 \pi )}^{2n}} \sum \limits_{k=1}^{\infty} \frac{\cos(2 \pi k \cdot x)}{k^{2n}} ，代入 n=1 ，得到

{\tilde{B}}_2 (x) = {(-1)}^{1+1} \frac{2 (2 \cdot 1)!}{{(2 \pi )}^{2 \cdot 1}} \sum \limits_{k=1}^{\infty} \frac{\cos(2 \pi k \cdot x)}{k^{2 \cdot 1}}

= \frac{1}{{\pi}^2} \sum \limits_{k=1}^{\infty} \frac{\cos(2 \pi k \cdot x)}{k^{2}} 。於是

{\tilde{B}}_2 (\frac{1}{2}) = B_2 (\frac{1}{2} = (x^2 - x + \frac{1}{6}) |_{x=\frac{1}{2}} = - \frac{1}{12}

= \frac{1}{{\pi}^2} \sum \limits_{k=1}^{\infty} \frac{ {(-1)}^{k} }{k^2}

= \frac{1}{{\pi}^2} \left[ - \sum \limits_{k=1}^{\infty} \frac{1}{ {(2k-1)}^2 } + \sum \limits_{k=1}^{\infty} \frac{1}{ {(2k)}^2 } \right]

= \frac{1}{{\pi}^2} \left[ - \frac{3}{4} \sum \limits_{k=1}^{\infty} \frac{1}{ {(k)}^2 } + \frac{1}{4} \sum \limits_{k=1}^{\infty} \frac{1}{ {(k)}^2 } \right]

= \frac{1}{{\pi}^2} \left[- \frac{1}{2} \sum \limits_{k=1}^{\infty} \frac{1}{ {(k)}^2 } \right]

= - \frac{1}{2 {\pi}^2} \frac{1}{2} \sum \limits_{k=1}^{\infty} \frac{1}{ {k}^2 } ，『容易』知道

\sum \limits_{k=1}^{\infty} \frac{1}{ {k}^2 } = \frac{{\pi}^2}{6} 矣。

這樣就能說成『創新』了嗎？假使代入 x=0，可得『歐拉最好的勝利』
{\tilde{B}}_{2n} (0) = {(-1)}^{n+1} \frac{2 (2n)!}{{(2 \pi )}^{2n}} \sum \limits_{k=1}^{\infty} \frac{1}{k^{2n}}

，更清楚表明之乎！！

不知誰知其後又將有

Riemann zeta function

The Riemann zeta function or Euler–Riemann zeta function, ζ(s), is a function of a complex variable s that analytically continues the sum of the Dirichlet series

  \zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}

for when the real part of s is greater than 1. More general representations of ζ(s) for all s are given below. The Riemann zeta function plays a pivotal role in analytic number theory and has applications in physics, probability theory, and applied statistics.

As a function of a real variable, Leonhard Euler first introduced and studied it in the first half of the eighteenth century without using complex analysis, which was not available at the time. Bernhard Riemann‘s 1859 article “On the Number of Primes Less Than a Given Magnitude” extended the Euler definition to a complex variable, proved its meromorphic continuation and functional equation, and established a relation between its zeros and the distribution of prime numbers.[2]

The values of the Riemann zeta function at even positive integers were computed by Euler. The first of them, ζ(2), provides a solution to the Basel problem. In 1979 Apéry proved the irrationality of ζ(3). The values at negative integer points, also found by Euler, are rational numbers and play an important role in the theory of modular forms. Many generalizations of the Riemann zeta function, such as Dirichlet series, Dirichlet L-functions and L-functions, are known.

Definition

Bernhard Riemann’s article on the number of primes below a given magnitude.

The Riemann zeta function ζ(s) is a function of a complex variable s = σ + it. (The notation s, σ, and t is used traditionally in the study of the zeta function, following Riemann.)

The following infinite series converges for all complex numbers s with real part greater than 1, and defines ζ(s) in this case:

{\displaystyle \zeta (s)=\sum _{n=1}^{\infty }n^{-s}={\frac {1}{1^{s}}}+{\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}+\cdots \qquad \sigma =\operatorname {Re} (s)>1.}

It can also be defined by the integral

\zeta (s)={\frac {1}{\Gamma (s)}}\int _{0}^{\infty }{\frac {x^{s-1}}{e^{x}-1}}\mathrm {d} x

where Γ(s) is the gamma function.

The Riemann zeta function is defined as the analytic continuation of the function defined for σ > 1 by the sum of the preceding series.

Leonhard Euler considered the above series in 1740 for positive integer values of s, and later Chebyshev extended the definition to Re(s) > 1.[3]

The above series is a prototypical Dirichlet series that converges absolutely to an analytic function for s such that σ > 1 and diverges for all other values of s. Riemann showed that the function defined by the series on the half-plane of convergence can be continued analytically to all complex values s ≠ 1. For s = 1 the series is the harmonic series which diverges to +∞, and

  \lim _{s\to 1}(s-1)\zeta (s)=1.

Thus the Riemann zeta function is a meromorphic function on the whole complex s-plane, which is holomorphic everywhere except for a simple pole at s = 1 with residue 1.

 

！乃今人們常常談起

特殊函數

特殊函數是指一些具有特定性質的函數，一般有約定俗成的名稱和記號，例如伽瑪函數、貝索函數、菲涅耳積分等。它們在數學分析、泛函分析、物理研究、工程應用中有著舉足輕重的地位。許多特殊函數是微分方程的解或基本函數的積分，因此積分表中常常會出現特殊函數，特殊函數的定義中也經常會出現積分。傳統上對特殊函數的分析主要基於對其的數值展開基礎上。隨著電子計算的發展，這個領域內開創了新的研究方法。因為微分方程的對稱性在數學和物理中的重要性，特殊函數理論也與李群和李代數密切相關。

事實上，對於哪些函數屬於特殊函數，並沒有明確的規定。函數列表中列出了一些通常被認為的特殊函數。廣義上，基本超越函數（即指數函數、對數函數、非有理次冪的冪函數、雙曲函數、三角函數等周期函數）也稱為特殊函數。

 

的耶？？