## a new way of defining $$e$$ the Euler number

This is inspired by a post on Quora by Alon Amit. I found this approach of defining e, the Euler number more intuitive than the traditional way($$e=\lim_{n \to \infty} (1+\frac{1}{n})^n$$). In addition, it explains the importance of the Euler number very well.

First of all, we need a function that solves the elementary differential equation:

$$\frac{d f(x)}{d x} = f(x)$$

Of course, without specifying an initial value, this equation will have infinite number of solutions. Just for convenience, let’s specify:

$$f(0)=1$$

By Picard–Lindelöf theorem, our equation:

$$\begin{gather*} \frac{d f(x)}{d x} = f(x)\\f(0)=1 \end{gather*}$$

has one and only one solution.

Now, let’s see what kind of property this function $$f(x)$$ has.

## 另外一种定义欧拉数$$e$$的方法

$$e=\lim_{n \to \infty} (1+\frac{1}{n})^n$$

$$f’=f$$                         ……（1）

$$f(x+a)=c*f(x)$$                     ……（2）

$$f(x+a)=f(a)*f(x)$$

$$f(1)=exp(1)=1+1+\frac{1}{2!}+\frac{1}{3!}+…+\frac{1}{n!}+…$$

$$\begin{gather*} \frac{d f(x)}{d x} = f(x)\\f(0)=1 \end{gather*}$$

$$\frac{d^2I}{dt^2}=-cI$$，其中$$c$$为大于零的常数。

$$k^2*e^{k*t}=-c*e^{k*t}$$，因此，

$$k^2=-c\\k=\pm\sqrt{c}*i$$

The reason this number e is considered the “natural” base of exponential functions is that this function is its own derivative.

This natural exponential function is identical with its derivative. This is really the source of all the properties of the exponential function, and the basic reason for its importance in applications…