【转载】三种证明欧拉恒等式的方法（3 methods of proving Euler's Formula ）

【转载】三种证明欧拉恒等式的方法（3 methods of proving Euler’s Formula ）

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These proofs followling was original state in WIKI.
FROM:http://en.wikipedia.org/wiki/Euler_formula

Proofs

使用泰勒级数
Here is a proof of Euler’s formula using Taylor series expansions as well as basic facts about the powers of i:

and so on. The functions ex, cos(x) and sin(x) (assuming x is real) can be expressed using their Taylor expansions around zero:

For complex z we define each of these functions by the above series, replacing x with z. This is possible because the radius of convergence of each series is infinite. We then find that

The rearrangement of terms is justified because each series is absolutely convergent. Taking z= x to be a real number gives the original identity as Euler discovered it.

利用微积分
Define the (possibly complex) function f(x), of real variable x, as

Division by zero is precluded since the equation

implies that is never zero.

The derivative of f(x), according to the quotient rule, is:

Therefore, f(x) must be a constant function in x. Because f(0) is known, the constant that f(x) equals for all real x is also known. Thus,

Rearranging, it follows that

Q.E.D.

利用常微分方程
Define the function g(x) by

Considering that i is constant, the first and second derivatives of g(x) are

because i 2 = ?1 by definition. From this the following 2nd-order linear ordinary differential equation is constructed:

or

Being a 2nd-order differential equation, there are two linearly independent solutions that satisfy it:

Both cos(x) and sin(x) are real functions in which the 2nd derivative is identical to the negative of that function. Any linear combination of solutions to a homogeneous differential equation is also a solution. Then, in general, the solution to the differential equation is

for any constants A and B. But not all values of these two constants satisfy the known initial conditions for g(x):

.
However these same initial conditions (applied to the general solution) are

resulting in

and, finally,

Q.E.D.

来源： http://hotblood660.blog.163.com/blog/static/9828435920083131534099/