$e$ 的故事說不盡， Eli Maor 先生曾寫過，誰人知其能貫串微積分之歷史耶？？精彩動人難下筆，自然天真無法傳！！

Exponential function

In mathematics, an exponential function is a function of the form

f(x)=b^{x}\,

in which the input variable $x$ occurs as an exponent. A function of the form $f(x)=b^{x+c}$ , where $c$ is a constant, is also considered an exponential function and can be rewritten as $f(x)=ab^{x}$ , with $a=b^{c}$ .

As a functions of a real variable, exponential functions are uniquely characterized by the fact that the growth rate of such a function (i.e., its derivative) is directly proportional to the value of the function. The constant of proportionality of this relationship is the natural logarithm of the base $b$ :

D_{x}b^{x}=b^{x}\log _{e}b

The constant e ≈ 2.71828… is the unique base for which the constant of proportionality is 1, so that the function’s derivative is itself:

D_{x}e^{x}=e^{x}\times \log _{e}e=e^{x}

Since changing the base of the exponential function merely results in the appearance of an additional constant factor, it is computationally convenient to reduce the study of exponential functions in mathematical analysis to the study of this particular function, conventionally called the “natural exponential function”,^[1]^[2] or simply, “the exponential function” and denoted by

x\mapsto e^{x}

\exp(\cdot )

The exponential function satisfies the fundamental multiplicative identity

e^{x+y}=e^{x}e^{y}

, for all

x,y\in \mathbb {R}

(In fact, this identity extends to complex-valued exponents.) It can be shown that complete set of continuous, nonzero solutions of the functional equation $f(x+y)=f(x)f(y)$ are the exponential functions, $f:\mathbb {R} \to \mathbb {R} ,\ x\mapsto b^{x}$ , with $b>0$ .

The argument of the exponential function can be any real or complex number or even an entirely different kind of mathematical object (e.g., a matrix).

Its ubiquitous occurrence in pure and applied mathematics has led mathematician W. Rudin to opine that the exponential function is “the most important function in mathematics”.^[3] In applied settings, exponential functions model a relationship in which a constant change in the independent variable gives the same proportional change (i.e. percentage increase or decrease) in the dependent variable. Such a situation occurs widely in the natural and social sciences; thus, the exponential function also appears in variety of contexts within physics, chemistry, engineering, mathematical biology, and economics.

The graph of $y=e^{x}$ is upward-sloping, and increases faster as $x$ increases. The graph always lies above the $x$ -axis but can get arbitrarily close to it for negative $x$ ; thus, the $x$ -axis is a horizontal asymptote. The slope of the tangent to the graph at each point is equal to its $y$ -coordinate at that point, as implied by its derivative function (see above). Its inverse function is the natural logarithm, denoted $\log$ ,^[4] $\ln$ ,^[5] or $\log _{e}$ ; because of this, some old texts^[6] refer to the exponential function as the antilogarithm.

The red curve is the exponential function. The black horizontal lines show where it crosses the green vertical lines.

內涵豐富七彩妝，形式多變性質好︰

Formal definition

Main article: Characterizations of the exponential function

The exponential function (in blue), and the sum of the first $n + 1$ terms of the power series on the left (in red).

The exponential function $\exp :\mathbb {C} \to \mathbb {C}$ can be characterized in a variety of equivalent ways. Most commonly, it is defined by the following power series:^[3]

\exp(z)=\sum _{k=0}^{\infty }{z^{k} \over k!}=1+z+{z^{2} \over 2}+{z^{3} \over 6}+{z^{4} \over 24}+\cdots

Since the radius of convergence of this power series is infinite, this definition is applicable to all complex numbers $z$ . The constant e is then defined as ${\textstyle e=\exp(1)=\sum _{k=0}^{\infty }(1/k!)}$ .

Less commonly, the real exponential function is defined as the solution $y$ to the equation

x=\int _{1}^{y}{1 \over t}\mathrm {d} t

The exponential function can also be defined as the following limit:^[7]

e^{x}=\lim _{n\rightarrow \infty }\left(1+{\frac {x}{n}}\right)^{n}

身處定點恆不動︰

Derivatives and differential equations

The derivative of the exponential function is equal to the value of the function. From any point $P$ on the curve (blue), let a tangent line (red), and a vertical line (green) with height $h$ be drawn, forming a right triangle with a base $b$ on the $x$ -axis. Since the slope of the red tangent line (the derivative) at $P$ is equal to the ratio of the triangle’s height to the triangle’s base (rise over run), and the derivative is equal to the value of the function, $h$ must be equal to the ratio of $h$ to $b$ . Therefore, the base b must always be 1.

The importance of the exponential function in mathematics and the sciences stems mainly from properties of its derivative. In particular,

{\frac {\mathrm {d} }{\mathrm {d} x}}e^{x}=e^{x}

Proof:

{\begin{aligned}e^{x}&=1+x+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+{\frac {x^{4}}{4!}}+{\frac {x^{5}}{5!}}+\cdots \\{\mathrm {d} \over \mathrm {d} x}e^{x}&={\mathrm {d} \over \mathrm {d} x}\left(1+x+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+{\frac {x^{4}}{4!}}+{\frac {x^{5}}{5!}}+\cdots \right)\\&=0+1+{\frac {2x}{2!}}+{\frac {3x^{2}}{3!}}+{\frac {4x^{3}}{4!}}+{\frac {5x^{4}}{5!}}+\cdots \\&=1+x+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+{\frac {x^{4}}{4!}}+{\frac {x^{5}}{5!}}+\cdots \\&=e^{x}\\\end{aligned}}

That is, $e x$ is its own derivative and hence is a simple example of a Pfaffian function. Functions of the form $ce x$ for constant $c$ are the only functions with that property (by the Picard–Lindelöf theorem). Other ways of saying the same thing include:

The slope of the graph at any point is the height of the function at that point.
The rate of increase of the function at $x$ is equal to the value of the function at $x$ .
The function solves the differential equation $y' = y$ .
$exp$ is a fixed point of derivative as a functional.

If a variable’s growth or decay rate is proportional to its size—as is the case in unlimited population growth (see Malthusian catastrophe), continuously compounded interest, or radioactive decay—then the variable can be written as a constant times an exponential function of time. Explicitly for any real constant $k$ , a function $f : R \to R$ satisfies $f' = kf$ if and only if $f (x) = ce kx$ for some constant $c$ .

Furthermore, for any differentiable function $f (x)$ , we find, by the chain rule:

{\mathrm {d} \over \mathrm {d} x}e^{f(x)}=f'(x)e^{f(x)}

何方仙女招之來？？！！

$\frac{d}{dx} e^{ln x} = \frac{d}{dx} x = 1$

$\therefore \frac{d}{dx} ln(x) = \frac{1}{x}$

加減乘除變換出︰

那麼『 泛函數方程式』有什麼『重要性』的嗎？就讓我們從一個『例子』開始談吧︰

$f(x + y) = f(x) \cdot f(y)$

。就是說有個『未知函數』 $f$ 將『定義域』中『兩數的加法』 $x + y$ 轉換成了『對應域』裡『兩數之乘法』 $f(x) \cdot f(y)$ 。這個『未知函數』的『整體性質』就由這個『方程式』來決定，比方講，這個函數 $f(0) = 1$ ，為什麼呢？因為 $f(0) = f(0 + 0) = f(0) \cdot f(0) = {f(0)}^2$ ，所以 $f(0) = 0$ 或 $f(0) = 1$ ，又為什麼不取 $f(0) = 0$ 的呢？因為如果取 $f(0) = 0$ 的話， $f(x) = f(x + 0) = f(x) \cdot f(0) = 0$ 只是個『零函數』罷了，一般叫做『平凡解』 trivial solution，雖然它為了『完整性』不能夠『被省略』，然而這個解『太顯然』的了 ── $0 + 0 = 0 \cdot 0$ ，似乎不必『再說明』的吧！不過有時這種『論證』的傳統給『初學者』帶來了『理解』的『困難』，故此特別指出。要怎麼『求解』泛函數方程式的呢？一般說來『非常困難』，有時甚至可能無法得知它到底是有『一個解』、『多個解』、『無窮解』或者根本就『無解』！！近年來漸漸的產生了一些『常用方法』，比方講『動態規劃』 Dynamic programming 中使用『分割征服』的『連續逼近法』或是將原方程式『變換』成適合『定點迭代法』等等。在此我們將介紹『無窮小分析』的『應用』，談一點對於一個『平滑函數』 $f(x)$ 來說，可以如何『思考』這種方程式的呢？由於 $f(x)$ 是『平滑的』，因此 $f(x)$ 它『可微分』，『導數』 $f^{\prime}(x)$ 存在而且『連續』，那麼 $f(x + \delta x)$ 就可以表示為

$f(x + \delta x) = f(x) + f^{\prime}(x) \cdot \delta x + \epsilon \cdot \delta x$

，此處 $\delta x \approx 0 \Longrightarrow \epsilon \approx 0$ 。由於 $f(x + y) = f(x) \cdot f(y)$ ，可得 $f(x + \delta x) = f(x) \cdot f(\delta x)$ ，因此 $f(x + \delta x) = f(x) \cdot f(\delta x) = f(x) + f^{\prime}(x) \cdot \delta x + \epsilon \cdot \delta x$ ，所以

$\frac{f^{\prime}(x)}{f(x)} = \frac{f(\delta x) - 1}{\delta x} - \epsilon$

。因為 $f(x)$ 是『平滑的』，那麼 $\frac{f^{\prime}(x)}{f(x)}$ 將存在且連續，然而 $\frac{f(\delta x) - 1}{\delta x} - \epsilon$ 與『變數』 $x$ 無關，於是當 $\delta x \approx 0$ 時，必定是某個『常數』 constant ，將之命作 $k$ ，如此那個『 泛函數方程式』就被改寫成了『微分方程式』

$f^{\prime}(x) = k \cdot f(x), \ f(0) = 1$

。它的『解』是 $f(x) = e^{k \cdot x}, \ k \neq 0$ ，為什麼 $k \neq 0$ 的呢？因為此時 $f^{\prime}(x) = 0$ ，將會得到 $f(x)$ 是個『常數函數』，這就是前面說的那個『平凡解』的啊！！

乍一看，這個『解法』果真『奇妙』，竟然變成了『微分方程式』的了！假使細察 $f(x + y) = f(x) \cdot f(y)$ 可說是『函數』 $f(x)$ 的『大域性質』，但是 $f(x + \delta x) = f(x) + f^{\prime}(x) \cdot \delta x + \epsilon \cdot \delta x$ 卻描述『函數』 $f(x)$ 之『微觀鄰域』，將此兩者合併考慮，果真是『可以的』嗎？如果說一個函數的『大域性質』自然『含括』 $\Longrightarrow$ 了它的『微觀鄰域』之『描述』，由於它是一個『平滑的』函數，那麼『微觀鄰域』的『微分方程式』難道不能『整合』 integrate $\Longrightarrow$ 成那個『大域性質』的嗎？

………

之前我們曾用『均值定理』

一個實數函數 $f$ 在閉區間 $[a, b]$ 裡『連續』且於開區間 $[a, b]$ 中『可微分』，那麼一定存在一點 $c, \ a < c < b$ 使得此點的『切線斜率』等於兩端點間的『割線斜率』，即 $f^{\prime}(c) = \frac{f(b) - f(a)}{b - a}$ 。

論證了『劉維爾定理』。這個『均值定理』的重要性在於，它將一個『連續』而且『可微分』的『函數』的『區間端點割線』與『區間內切線』聯繫了起來，使我們可以『確定』一個『等式』的『存在』。就讓我們再舉一個『對數性函數』 $f(x \cdot y) = f(x) + f(y)$ 的例子，看看它的『運用』吧。首先 $f(1) = f(1 \cdot 1) = f(1) + f(1) \Longrightarrow f(1) = 0$ ，其次 $f(x \cdot \frac{1}{x}) = f(1) = 0 = f(x) + f(\frac{1}{x}) \Longrightarrow f(\frac{1}{x}) = - f(x)$ ，所以 $f(\frac{x}{y}) = f(x \cdot \frac{1}{y}) = f(x) + f(\frac{1}{y}) = f(x) -f(y)$ 。因此

$f(x + \delta x) - f(x) = f(\frac{x + \delta x}{\delta x}) = f(1 + \frac{\delta x}{ x})$

$= f^{\prime}(\eta) \left[(1 + \frac{\delta x}{x}) - 1 \right], \ \eta \in (1, 1 + \delta x)$

$= f^{\prime}(\eta) \frac{\delta x}{x}$

，為什麼呢？因為 $f(x)$ 在『閉區間』 $[1, 1 + \delta x]$ 是『平滑的』，按照『均值定理』，存在一個 $\eta \in (1, 1+ \delta x)$ 使得

$f^{\prime}(\eta) = \frac{f( 1 + \frac{\delta x}{ x}) - f(1)}{(1 + \frac{\delta x}{x}) - 1} = \frac{f( 1 + \frac{\delta x}{ x})}{ \frac{\delta x}{x}}$ 。

$\therefore f(x + \delta x) = f(x) + f^{\prime}(\eta) \frac{\delta x}{x} = f(x) + f^{\prime}(x) \cdot \delta x + \epsilon \cdot \delta x$ ，於是我們可以得到

$f^{\prime}(x) = \frac{f^{\prime}(\eta)}{x} - \epsilon$ ，也就是說『函數』 $f(x)$ 滿足

$f^{\prime}(x) = \frac{k}{x} , \ f(1)= 0, \ k= f^{\prime}(1)$ 。

它的『解』果真就是 $f(x) = k \ln{(x)}$ 的啊！！

─── 摘自《【Sonic π】電聲學之電路學《四》之《一》》

整體鄰域特徵在！！？？

雙曲函數隨之起︰

雙曲函數

在數學中，雙曲函數是一類與常見的三角函數（也叫圓函數）類似的函數。最基本的雙曲函數是雙曲正弦函數 $\sinh$ 和雙曲餘弦函數 $\cosh$ ，從它們可以導出雙曲正切函數 $\tanh$ 等，其推導也類似於三角函數的推導。雙曲函數的反函數稱為反雙曲函數。

雙曲函數的定義域是實數，其自變量的值叫做雙曲角。雙曲函數出現於某些重要的線性微分方程的解中，譬如說定義懸鏈線和拉普拉斯方程。

基本定義

sinh, cosh和tanh

csch, sech和coth

$\sinh x={{e^{x}-e^{-x}} \over 2}$
$\cosh x={{e^{x}+e^{-x}} \over 2}$
$\tanh x={{\sinh x} \over {\cosh x}}$
$\coth x={1 \over {\tanh x}}$
${\mathop {\rm {sech}}}x={1 \over {\cosh x}}$
${\mathop {\rm {csch}}}x={1 \over {\sinh x}}$

函數 $\cosh x\!$ 是關於y軸對稱的偶函數。函數 $\sinh x\!$ 是奇函數。

如同當 $t$ 遍歷實數集 $\mathbb {R}$ 時，點（ $\cos t\!$ , $\sin t\!$ ）的軌跡是一個圓 $x^{2}+y^{2}=1$ 一樣，當 $t$ 遍歷實數集 $\mathbb {R}$ 時，點（ $\cosh t\!$ , $\sinh t\!$ ）的軌跡是單位雙曲線 $x^{2}-y^{2}=1$ 的右半邊。這是因為有以下的恆等式：

\cosh ^{2}t-\sinh ^{2}t=1\,

參數t不是圓角而是雙曲角，它表示在x軸和連接原點和雙曲線上的點（ $\cosh t\!$ , $\sinh t\!$ ）的直線之間的面積的兩倍。

歷史

在直角雙曲線(方程y = 1/x)下，雙曲線三角形(黃色)，和對應於雙曲角u的雙曲線扇形(紅色)。這個三角形的邊分別是雙曲函數中cosh和sinh的√2倍。

在18世紀，約翰·海因里希·蘭伯特介入了雙曲函數^[1]，並計算了雙曲幾何中雙曲三角形的面積^[2]。自然對數函數是在直角雙曲線 $xy=1$ 下定義的，可構造雙曲線直角三角形，底邊在線 $y=x$ 上，一個頂點是原點，另一個頂點在雙曲線。這裡以自然對數即雙曲角作為參數的函數，是自然對數的逆函數指數函數，即要形成指定雙曲角u，在漸進線即x或y軸上需要有的x或y的值。顯見這裡的底邊是 $\left(e^{u}+e^{-u}\right){\frac {\sqrt {2}}{2}}$ ，垂線是 $\left(e^{u}-e^{-u}\right){\frac {\sqrt {2}}{2}}$ 。