Euler's formula in the Complex Variable Function Theory

E ^ IX = cosx + is, e is the base of the natural logarithm, And I is the unit of virtual number. It expands the definite domain of trigonometric functions to the plural, establishes the relationship between trigonometric functions and exponential functions, and plays an important role in the theory of complex functions.

E ^ IX = proof of cosx + ISX:

Because e ^ x = 1 + x/1! + X ^ 2/2! + X ^ 3/3! + X ^ 4/4! + ......

Cos x = 2/2 ^! + X ^ 4/4! -X ^ 6/6 !......

SiN x = x-x ^ 3/3! + X ^ 5/5! -X ^ 7/7 !......

In the expansion of e ^ X, replace X with ± IX. (± I) ^ 2 =-1, (± I) ^ 3 = ∓ I, (± I) ^ 4 = 1 ......

E ^ ± IX = 1 ± x/1! -X ^ 2/2! + X ^ 3/3! Listen x ^ 4/4 !......

= (2/2 ^! + ......) ± I (X-x ^ 3/3 !......)

So e ^ ± IX = cosx ± isinx

Replace X in the formula with-X:

E ^-Ix = cosx-isinx, and then use the two-phase addition and subtraction method to obtain:

SiNx = (E ^ IX-e ^-IX)/(2I), cosx = (E ^ IX + e ^-IX)/2. These two are also called Euler's formulas. Take the X in E ^ IX = cosx + ISX as π to get it:

E ^ I π + 1 = 0.

This constant equation is also called the Euler's formula. It is the most fascinating formula in mathematics. It associates the most important digits in mathematics: Two super numbers: the bottom e of the natural logarithm, circumference rate π, two units: Virtual number unit I and natural number Unit 1, and common mathematics 0. Mathematicians think of it as a "formula created by God". We can only look at it, not understand it.