아름다움 중에서도 아름다운 Euler's identity 오일러의 공식

Euler's identity 
 

e is Euler's number, the base of natural logarithms,

i is the imaginary unit, which satisfies i² = −1, and

     π is pi, the ratio of the circumference of a circle to its diameter.

삼각함수와 지수함수에 대한 관계를 나타낸다. 오일러의 등식은 이 공식의 특수한 경우이다.

오일러의 공식은 가장 아름다운 것들을 포함하고 있다. 세가지 기본적 산술연산: 덧셈, 곱셈, 지수를 포함하고 있으며 다섯가지의 가장 기본적인 상수: 0, 1, pi, theta, i 를 포함하고 있다.

The number 0, the additive identity.

The number 1, the multiplicative identity.

The number π, which is ubiquitous in trigonometry, the geometry of Euclidean space, and analytical mathematics (π = 3.14159265...)

The number e, the base of natural logarithms, which occurs widely in mathematical and scientific analysis (e = 2.718281828...). Both π and e are transcendental numbers.

The number i, the imaginary unit of the complex numbers, a field of numbers that contains the roots of all polynomials (that are not constants), and whose study leads to deeper insights into many areas of algebra andcalculus, such as integration in calculus.
 

The identity is a special case of Euler's formula fromcomplex analysis, which states that

for any real number x. (Note that the arguments to thetrigonometric functions sine and cosine are taken to be in radians, and not in degrees.) In particular, with x = π, or one half turn around the circle:

Since

and

it follows that

which gives the identity

 

 

The exponential function e^z can be defined as the limit of(1 + z/N)^N, as N approaches infinity, and thus e^iπ is the limit of(1 +iπ/N)^N. In this animation N takes various increasing values from 1 to 100. The computation of (1 + iπ/N)^N is displayed as the combined effect of N repeated multiplications in the complex plane, with the final point being the actual value of (1 +iπ/N)^N. It can be seen that as N gets larger (1 +iπ/N)^N approaches a limit of −1.

http://en.wikipedia.org/wiki/Euler's_identity
http://ko.wikipedia.org/wiki/%EC%98%A4%EC%9D%BC%EB%9F%AC%EC%9D%98_%EA%B3%B5%EC%8B%9D