如果(e^x)'=e^x,那么链式法则给出[e^(ax)]'=a e^x.如果让a=jw,j是-1的平方根，那就能联上三角函数。因为 [cos(wt)+jsin(wt)]'=jw[cos(wt)+jsin(wt)]！也就是说cos(wt)+jsin(wt)=e^(jwt)！如果让wt=\pi那么就有-1=e^(j\pi)或e^(j\pi)+1=0,很多妙事。。。。
这些妙事都基于有个函数其导数是自己，它是求导运算的特征函数.一个量的变化率等于这个量本身的大小，可想见其会在物理世界有用，因为物体常常只能根据自己量的大小来确定其变化---说糊涂了吧？像哲学了吧？：）
顺便说说，\pi重要还是因为其为转一圈的角度，而不仅是周长直径比。所有周期运动都是转圈圈（好多是不园的）。如果不是这么多周期运动则我们的世界太难预料，生命难以演化，也就没有新语丝了：）胡扯一通了对不起。但以上看出e在描述转圈圈时有大作用，所以重要。所以e和\pi双宿双飞，两峰并秀。
Compound Interest [edit]
Let us suppose that we deposit an amount A0 in the bank on New Year's Day, and furthermore that every year on the year the amount is augmented by a rate r times the present amount. Then the amount A in the bank on any given New Year's Day, t years after the first is given by the expression
A=A_0(1+r)^t\,\!.
Unfortunately, if we withdraw the money three days before the New Year, we don't get any of the interest payment for that year. A fairer system would involve calculating interest n times a year at the rate r / n. In fact this gives us a slightly different value even if we take our money out on a New Year's Day, because every time we calculate interest, we receive interest on our previous interest. The amount A we receive with this improved system is given by the expression
A=A_0(1+\frac{r}{n})^{nt}.
With this flexible system, we could set n to 12 to compound every month, or to 365 to compound every day or to about 31536000 to compound every second. But why stop there? Why not compound the interest every moment? What is really meant by that is this: as we increase n does the value for A get ever greater with n or does it approach some reasonable quantity? If the latter is the case, then it is meaningful to ask, "What does A approach?" As we can see from the following table with sample values, this is in fact the case.
n A
1 1.02500
12 1.02529
365 1.02531
31536000 1.02532
100000000 1.02532
A0 = 1, r = .025, t = 1
As we can see, as n goes off toward infinity, A approaches a finite value. Taking this to heart, we may come to our final system in which we define A as follows:
A=\lim_{n\rightarrow \infty}A_0(1+\frac{r}{n})^{nt}.
Thus we set A now not to A_0(1+\frac{r}{n})^{nt} evaluated for some large n, but rather to the limit of that value as n approaches infinity. This is the formula for continually compounded interest. To clean up this formula, note that neither A0 nor t "interfere" in any way with the evaluation of the limit, and may consequently be moved outside of the limit without affecting the value of the expression:
A=A_0B^t\,\!,
where
B=\lim_{n\rightarrow \infty}(1+\frac{r}{n})^n.
We can see from the form of the expression that A increases exponentially with t much as it did in our very first equation. The difference is that the original base (1 + r) has been replaced with the base B which we have yet to simplify.
Take a moment to step back and do the following exercises:
1. Without looking back, see if you can write down the expressions that represent
* yearly interest
* semiannual interest
* monthly interest
* interest n times a year
* continually compounded interest
2. Think about how much money you have. Figure out how long you would have to leave your money in a bank that compounds interest monthly before you became a millionaire, with a yearly interest rate of
* .02 (common for a savings account)
* .07 (average gain in the US stock market over a reasonably long period).