生成函數『形式』何其多？

Ordinary generating function

The ordinary generating function of a sequence a_n is

G(a_n;x)=\sum_{n=0}^\infty a_nx^n.

When the term generating function is used without qualification, it is usually taken to mean an ordinary generating function.

If a_n is the probability mass function of a discrete random variable, then its ordinary generating function is called a probability-generating function.

The ordinary generating function can be generalized to arrays with multiple indices. For example, the ordinary generating function of a two-dimensional array a_{m, n} (where n and m are natural numbers) is

G(a_{m,n};x,y)=\sum_{m,n=0}^\infty a_{m,n}x^my^n.

Exponential generating function

The exponential generating function of a sequence a_n is

\operatorname{EG}(a_n;x)=\sum _{n=0}^{\infty} a_n \frac{x^n}{n!}.

Exponential generating functions are generally more convenient than ordinary generating functions for combinatorial enumeration problems that involve labelled objects.^[3]

Poisson generating function

The Poisson generating function of a sequence a_n is

\operatorname{PG}(a_n;x)=\sum _{n=0}^{\infty} a_n e^{-x} \frac{x^n}{n!} = e^{-x}\, \operatorname{EG}(a_n;x).

Lambert series

The Lambert series of a sequence a_n is

\operatorname{LG}(a_n;x)=\sum _{n=1}^{\infty} a_n \frac{x^n}{1-x^n}.

Note that in a Lambert series the index n starts at 1, not at 0, as the first term would otherwise be undefined.

Bell series

The Bell series of a sequence a_n is an expression in terms of both an indeterminate x and a prime p and is given by^[4]

\operatorname{BG}_p(a_n;x)=\sum_{n=0}^\infty a_{p^n}x^n.

Dirichlet series generating functions

Formal Dirichlet series are often classified as generating functions, although they are not strictly formal power series. The Dirichlet series generating function of a sequence a_n is^[5]

\operatorname{DG}(a_n;s)=\sum _{n=1}^{\infty} \frac{a_n}{n^s}.

The Dirichlet series generating function is especially useful when a_n is a multiplicative function, in which case it has an Euler product expression^[6] in terms of the function’s Bell series

\operatorname{DG}(a_n;s)=\prod_{p} \operatorname{BG}_p(a_n;p^{-s})\,.

If a_n is a Dirichlet character then its Dirichlet series generating function is called a Dirichlet L-series.

Polynomial sequence generating functions

The idea of generating functions can be extended to sequences of other objects. Thus, for example, polynomial sequences of binomial type are generated by

e^{xf(t)}=\sum_{n=0}^\infty \frac{p_n(x)}{n!} t^n

where p_n(x) is a sequence of polynomials and f(t) is a function of a certain form. Sheffer sequences are generated in a similar way. See the main article generalized Appell polynomials for more information.

『變換』思維道其妙︰

Laplace transform

In mathematics the Laplace transform is an integral transform named after its discoverer Pierre-Simon Laplace (/ləˈplɑːs/). It takes a function of a positive real variable $t$ (often time) to a function of a complex variable $s$ (frequency).

The Laplace transform is very similar to the Fourier transform. While the Fourier transform of a function is a complex function of a real variable (frequency), the Laplace transform of a function is a complex function of a complex variable. Laplace transforms are usually restricted to functions of $t$ with $t > 0$ . A consequence of this restriction is that the Laplace transform of a function is a holomorphic function of the variable $s$ . Unlike the Fourier transform, the Laplace transform of a distribution is generally a well-behaved function. Also techniques of complex variables can be used directly to study Laplace transforms. As a holomorphic function, the Laplace transform has a power series representation. This power series expresses a function as a linear superposition of moments of the function. This perspective has applications in probability theory.

The Laplace transform is invertible on a large class of functions. The inverse Laplace transform takes a function of a complex variable s (often frequency) and yields a function of a real variable t (time). Given a simple mathematical or functional description of an input or output to a system, the Laplace transform provides an alternative functional description that often simplifies the process of analyzing the behavior of the system, or in synthesizing a new system based on a set of specifications.^[1] So, for example, Laplace transformation from the time domain to the frequency domain transforms differential equations into algebraic equations and convolution into multiplication. It has many applications in the sciences and technology.

『對數』概念始其用︰

一六一四年 John Napier 約翰‧納皮爾在一本名為《 Mirifici Logarithmorum Canonis Descriptio 》── 奇妙的對數規律的描述 ── 的書中，用了三十七頁解釋『對數』 log ，以及給了長達九十頁的對數表。這有什麼重要的嗎？想一想即使在今天用『鉛筆』和『紙』做大位數的加減乘除，尚且困難也很容易算錯，就可以知道對數的發明，對計算一事貢獻之大的了。如果用一對一對應的觀點來看，對數把『乘除』運算『變換』成『加減』運算︰

$\log {a * b} = \log{a} + \log{b}$

$\log {a / b} = \log{a} - \log{b}$

，更不要說還可以算『平方』、『立方』種種和開『平方根』、『立方根』等等的計算了。

$\log {a^n} = n * \log{a}$

傳聞納皮爾還發明了的『骨頭計算器』，他的書對於之後的天文學、力學、物理學、占星學的發展都有很大的影響。他的運算變換 Transform 的想法，開啟了『換個空間』解決『數學問題』的大門，比方『常微分方程式的 Laplace Transform』與『頻譜分析的傅立葉變換』等等。

這個對數畫起來是這個樣子︰

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不只如此這個對數關係竟然還跟人類之『五官』── 眼耳鼻舌身 ── 受到『刺激』── 色聲香味觸 ── 的『感覺』強弱大小有關。一七九五年出生的 Ernst Heinrich Weber 韋伯，一位德國物理學家，是一位心理物理學的先驅，他提出感覺之『方可分辨』JND just-noticeable difference 的特性。比方說你提了五公斤的水，再加上半公斤，可能感覺差不了多少，要是你沒提水，說不定會覺的突然拿著半公斤的水很重。也就是說在『既定的刺激』下，感覺的方可分辨性大小並不相同。韋伯實驗後歸結成一個關係式︰

ΔR/R = K

R: 既有刺激之物理量數值
ΔR: 方可分辨 JND 所需增加的刺激之物理量數值
K: 特定感官之常數，不同的感官不同

。之後 Gustav Theodor Fechner 費希納，一位韋伯派的學者，提出『知覺』perception 『連續性』假設，將韋伯關係式改寫為︰

$dP = k \frac {dS}{S}$

，求解微分方程式得到︰

$P = k \ln S + C$

假如刺激之物理量數值小於 $S_0$ 時，人感覺不到 $P = 0$ ，就可將上式寫成︰

$P = k \ln \frac {S}{S_0}$

這就是知名的韋伯-費希納定律，它講著：在絕對閾限 $S_0$ 之上，主觀知覺之強度的變化與刺激之物理量大小的改變呈現自然對數的關係，也可以說，如果刺激大小按著幾何級數倍增，所引起的感覺強度卻只依造算術級數累加。

其後有人將它應用到『行銷學』的領域︰

消費者對價格變化的感受大都取決於改變的百分比

，也就是說︰

十塊錢東西變成十五塊，叫『天價』的貴，
二十五塊錢東西變成三十塊，稱『坑人』的價，
一百塊錢東西變成一百零五塊，沒『感覺』的吧。

── 真是可憐的『小吃業者』，沒在怕的『頂級餐廳』！！──

─── 摘自《千江有水千江月》

歐拉公式 $e^{i \theta} = \cos(\theta) + i \sin(\theta)$ 開其門︰

$e^{i \pi} + 1 = 0$

一五七二年義大利數學家拉斐爾‧邦貝利 Rafael Bombelli 是文藝復興時期歐洲著名的工程師，也是一個卓越的數學家，出版了《代數學》 L’Algebra 一書，他在書中討論了『負數的平方根』 $\sqrt{- a}, \ a>0$ ，這在歐洲產生了廣泛影響力。

一六三七年笛卡爾在他的著作《幾何學》 La Géométrie 書中創造了『虛數』imaginary numbers 一詞，說明這種『真實上並不存在的數字』。

瑞士大數學家和物理學家李昂哈德‧尤拉 Leonhard Euler 傳說年輕時曾研讀神學，一生虔誠篤信上帝，並不能容忍任何詆毀上帝的言論在他面前發表。一回，德尼‧狄德羅 Denis Diderot ── 法國啟蒙思想家、唯物主義哲學家、無神論者和作家，百科全書派的代表 ── 造訪葉卡捷琳娜二世的宮廷，尤拉挑戰狄德羅說︰『先生， $e^{i \pi} + 1 = 0$ ，所以上帝存在，請回答！』。作者以為這或許只是個『杜撰』。然而尤拉是位多產的作家，一生著作有六十到八十巨冊。一七八三年九月十八日，晚餐後，尤拉邊喝著茶邊和小孫女玩耍，突然間，煙斗從他手中掉了下來。他說了聲：『我的煙斗』，將彎腰去撿，就再也沒有站起來了，他祇是抱著頭說了一句：『我死了』。法國哲學家馬奎斯‧孔多塞 marquis de Condorcet 講︰..il cessa de calculer et de vivre，『尤拉停止了計算和生命』！！

一七九七年挪威‧丹麥數學家卡斯帕爾‧韋塞爾 Caspar Wessel 在『Royal Danish Academy of Sciences and Letters』上發表了『Om directionens analytiske betegning』，提出了『複數平面』，研究了複數的幾何意義，由於是用『丹麥文』寫成的，幾乎沒有引起任何重視。一八零六年法國業餘數學家讓-羅貝爾‧阿爾岡 Jean-Robert Argand 與一八三一年德國著名大數學家约翰‧卡爾‧弗里德里希‧高斯 Johann Karl Friedrich Gauß 都再次『重新發現』同一結果！！

虛數軸和實數軸構成的平面稱作複數平面，複平面上每一點對應著一個複數。

那麽要怎樣理解『複數』 $z = x + i \ y$ 的呢？如果說『複數』起源於『方程式』的『求解』，比方說 $x^2 + 1 = 0, \ x = \pm i$ ，這定義了『 $i = \sqrt{-1}$ 』，但是它的『意義』依然晦澀。即使說從『複數平面』的每一個『點』 $(x, y)$ 都對應著一個『複數』 $z = x + i \ y$ 可能還是不清楚『 $i$ 』的意思到底是什麼？假使再從『複數』的『加法上看』︰

假使 $z_1 = x_1 + i \ y_1$ 、 $z_2 = x_2 + i \ y_2$ ，

那麼 $z_1 + z_2 = (x_1 + x_2) + i \ (y_1 + y_2)$ 。

這是一種類似『向量』的加法，是否『 $i$ 』的意義就藏在其中的呢？

─── 摘自《【Sonic π】電聲學補充《二》》

歷代耕耘方法傳。

History

The Laplace transform is named after mathematician and astronomer Pierre-Simon Laplace, who used a similar transform (now called the z-transform) in his work on probability theory.^[2] The current widespread use of the transform (mainly in engineering) came about during and soon after World War II ^[3] although it had been used in the 19th century by Abel, Lerch, Heaviside, and Bromwich.

The early history of methods having some similarity to Laplace transform is as follows. From 1744, Leonhard Euler investigated integrals of the form

z=\int X(x)e^{ax}\,dx\quad {\text{ and }}\quad z=\int X(x)x^{A}\,dx

as solutions of differential equations but did not pursue the matter very far.^[4]

Joseph Louis Lagrange was an admirer of Euler and, in his work on integrating probability density functions, investigated expressions of the form

\int X(x)e^{-ax}a^{x}\,dx,

which some modern historians have interpreted within modern Laplace transform theory.^[5]^[6]^{[clarification needed]}

These types of integrals seem first to have attracted Laplace’s attention in 1782 where he was following in the spirit of Euler in using the integrals themselves as solutions of equations.^[7] However, in 1785, Laplace took the critical step forward when, rather than just looking for a solution in the form of an integral, he started to apply the transforms in the sense that was later to become popular. He used an integral of the form

\int x^{s}\phi (x)\,dx,

akin to a Mellin transform, to transform the whole of a difference equation, in order to look for solutions of the transformed equation. He then went on to apply the Laplace transform in the same way and started to derive some of its properties, beginning to appreciate its potential power.^[8]

Laplace also recognised that Joseph Fourier‘s method of Fourier series for solving the diffusion equation could only apply to a limited region of space because those solutions were periodic. In 1809, Laplace applied his transform to find solutions that diffused indefinitely in space.^[9]

莫說『形式』無『實質』︰

Formal definition

The Laplace transform is a frequency-domain approach for continuous time signals irrespective of whether the system is stable or unstable. The Laplace transform of a function $f (t)$ , defined for all real numbers $t \geq 0$ , is the function $F (s)$ , which is a unilateral transform defined by

F(s)=\int _{0}^{\infty }f(t)e^{-st}\,dt

where s is a complex number frequency parameter

s=\sigma +i\omega

, with real numbers

σ

and

ω

Other notations for the Laplace transform include $L {f}$ , or alternatively $L {f (t)}$ instead of $F$ .

The meaning of the integral depends on types of functions of interest. A necessary condition for existence of the integral is that $f$ must be locally integrable on $[0, \infty)$ . For locally integrable functions that decay at infinity or are of exponential type, the integral can be understood to be a (proper) Lebesgue integral. However, for many applications it is necessary to regard it to be a conditionally convergent improper integral at $\infty$ . Still more generally, the integral can be understood in a weak sense, and this is dealt with below.

One can define the Laplace transform of a finite Borel measure $μ$ by the Lebesgue integral^[10]

{\mathcal {L}}\{\mu \}(s)=\int _{[0,\infty )}e^{-st}\,d\mu (t).

An important special case is where $μ$ is a probability measure, for example, the Dirac delta function. In operational calculus, the Laplace transform of a measure is often treated as though the measure came from a probability density function $f$ . In that case, to avoid potential confusion, one often writes

{\mathcal {L}}\{f\}(s)=\int _{0^{-}}^{\infty }f(t)e^{-st}\,dt,

where the lower limit of $0 -$ is shorthand notation for

\lim _{\varepsilon \downarrow 0}\int _{-\varepsilon }^{\infty }.

This limit emphasizes that any point mass located at $0$ is entirely captured by the Laplace transform. Although with the Lebesgue integral, it is not necessary to take such a limit, it does appear more naturally in connection with the Laplace–Stieltjes transform.

總綱立論『心要』在︰

Probability theory

In pure and applied probability, the Laplace transform is defined as an expected value. If $X$ is a random variable with probability density function $f$ , then the Laplace transform of $f$ is given by the expectation

{\mathcal {L}}\{f\}(s)=E\!\left[e^{-sX}\right]\!.

By abuse of language, this is referred to as the Laplace transform of the random variable $X$ itself. Replacing $s$ by $- t$ gives the moment generating function of $X$ . The Laplace transform has applications throughout probability theory, including first passage times of stochastic processes such as Markov chains, and renewal theory.

Of particular use is the ability to recover the cumulative distribution function of a continuous random variable $X$ by means of the Laplace transform as follows^[11]

F_{X}(x)={\mathcal {L}}^{-1}\!\left\{{\frac {1}{s}}E\left[e^{-sX}\right]\right\}\!(x)={\mathcal {L}}^{-1}\!\left\{{\frac {1}{s}}{\mathcal {L}}\{f\}(s)\right\}\!(x).

觀其『會通』理念純︰

Relation to power series

The Laplace transform can be viewed as a continuous analogue of a power series. If $a (n)$ is a discrete function of a positive integer $n$ , then the power series associated to $a (n)$ is the series

\sum _{n=0}^{\infty }a(n)x^{n}

where $x$ is a real variable (see Z transform). Replacing summation over $n$ with integration over $t$ , a continuous version of the power series becomes

\int _{0}^{\infty }f(t)x^{t}\,dt

where the discrete function $a (n)$ is replaced by the continuous one $f (t)$ . (See Mellin transform below.)

Changing the base of the power from $x$ to $e$ gives

\int _{0}^{\infty }f(t)\left(e^{\ln {x}}\right)^{t}\,dt

For this to converge for, say, all bounded functions $f$ , it is necessary to require that $ln x < 0$ . Making the substitution $- s = ln x$ gives just the Laplace transform:

\int _{0}^{\infty }f(t)e^{-st}\,dt

In other words, the Laplace transform is a continuous analog of a power series in which the discrete parameter $n$ is replaced by the continuous parameter $t$ , and $x$ is replaced by $e - s$ .

『出入自然』自為功！

s-domain equivalent circuits and impedances

The Laplace transform is often used in circuit analysis, and simple conversions to the $s$ -domain of circuit elements can be made. Circuit elements can be transformed into impedances, very similar to phasor impedances.

Here is a summary of equivalents:

Note that the resistor is exactly the same in the time domain and the $s$ -domain. The sources are put in if there are initial conditions on the circuit elements. For example, if a capacitor has an initial voltage across it, or if the inductor has an initial current through it, the sources inserted in the $s$ -domain account for that.

The equivalents for current and voltage sources are simply derived from the transformations in the table above.

Example 1: Solving a differential equation

In nuclear physics, the following fundamental relationship governs radioactive decay: the number of radioactive atoms $N$ in a sample of a radioactive isotope decays at a rate proportional to $N$ . This leads to the first order linear differential equation