Who can prove the e x s used in the proof of Euler s formula e ix cosx isinx

The three formulas are Taylor series, college calculus or advanced mathematics to learn, these three formulas are very basic, science and engineering students must memorize in college, you want to understand can (Taylor series), the information and derivation must be very complete.

Euler's formula e(ix)=cosx+isinx is just a definition, there is no derivation, you can think of f(ix)=cosx+isinx; And this f(ix) is very clever, very similar to the nature of e x as we know, (e.g. f(ix)*e x=f(ix+x)) and thus written e(ix), but in fact it is not the traditional e x, just a way of writing it. e (i) +1 = 0 is the case of x=pi of this definition, see "Complex Variable Functions" for details, which is also a university course.

To add: define the function f(z)=e x(cosy+isiny) in the complex plane, which has certain properties similar to e x, such as f'(z)=f(z).Write f(z) as expz, that is, expz=e x(cosy+isiny), for convenience, often use e z instead of expz, write e z=e x (cosy+isiny), here e z has no power meaning, only the meaning of symbols, and z=ix and i*pi are your two formulas (supplementary content from the "complex variable function" published by Xi'an Jiaotong University).

There is a very straightforward but not rigorous proof that if you know a function and there is another function that is exactly the same as it is, no matter how many derivatives it has, then the two functions must be the same.

And the series of coefficients that Taylor appears is actually obtained by the number 1 after multiple integrations. And the power of x is calculated with the integral.

Why choose the power of x, because he can naturally twist into various traits. Try drawing (x+a) +x+b) +x+c)=0 with geogebra (search directly on the official website), and by adjusting the parameter abc you can twist it up and down and pan it anywhere. The same principle applies to high-power power functions.

Now let's assume that we don't need these two functions to be exactly the same, and let's ditch both ends. If the accuracy is satisfied in the seventh derivative, then we set the value of the seventh derivative into it, and then integrate it once to find the sixth order, at this time the seventh order has already appeared x, continue to integrate until the integration is all behind"Derivative value"*(x^n)/(n!)

I don't know how much you've learned, so I won't go into detail about derivatives and integrals, but I'll take a look at the encyclopedia?

As for e ix, according to e ix=cosx+isinx, we can know that this is actually a complex number. A complex number is actually a vector. When we add something to this index, the magic happens.

Adding the real number we know that e (ix+a) = (e a)*(e ix), which is equivalent to stretching the vector, and the length of the vector depends on how much you add to it. Adding complex numbers is actually increasing x, and I still recommend drawing a picture with geogebra for sophomores, and you will find that it is spinning in circles.

The addition of vectors is the end of the end, that is, the superposition of two vectors that can be rotated in circles and the size is controllable. We can use this to draw any graph (refer to the Fourier transform plot at station B). The Fourier transform is essentially a vector representation of any function.

Let's consider the one-dimensional case again, a set of vectors end to end together and turn circles (note that they are circles at the same time), it is impossible to draw sharp corners, that is to say, we can draw a waveform, so let's record some sounds, write his equations, and then modify them, do you think of anything? We can make a sound.

Okay, let's think about three dimensions, one latitude is time, and two dimensions are space, which is very abstract and requires a bit of brainstorming. It's a two-dimensional wave walking. We extend it to four dimensions and give it meaning, two electric and magnetic fields that are perpendicular to each other and walk in four dimensions of space-time, this is electromagnetic waves!

That's a quantum! Popular science books explain quantum mechanics when the wave equation pops up, among them"Volatility"That's what it means.

I believe this will give you a further understanding of E ix, and most importantly know how much this equation and its implications have on this world.

Euler's formulae(ix)=cosx+isinx is just a definition, there is no derivation, you can think of f(ix)=cosx+isinx; And this age is very clever without f(ix), which is very similar to the nature of e x that we know, (e.g. f(ix)*e x=f(ix+x)) and thus writes e (ix), but in fact it is not the traditional e x, just a way of writing.

Derivation process: because cosx+isinx=e ix

cosx-isinx=e^-ix

Add the two formulas to get: 2cosx=e ix+e -ix, and divide 2 to get cosx=(e ix+e -ix) 2.

Subtract the two formulas to get: 2isinx=e ix-e -ix, and divide 2i to get sinx=(e ix-e -ix) 2i.

Meaning. The identity, also known as Euler's formula, is one of the most fascinating formulas in mathematics that connects the most important numbers in mathematics: the two transcendental numbers: the base of the natural logarithm.

e, pi.

Two units: the imaginary unit i and the natural number.

Unit 1; and 0, which has been called one of the great discoveries of mankind. Mathematicians describe it as "God's created formula."

<> Euler's formula and Euler's square mill first Cheng Qin Hall pushes the blind limb guide.

e ix=cosx+isinx, e is the bottom of the self-hungry logarithm, i is the imaginary unit.

Replace x with -x in the formula to get :

e -ix=cosx-isinx, and then use the method of addition and subtraction of the two formulas to obtain: pre-stupid.

sinx=(e^ix-e^-ix)/(2i),cosx=(e^ix+e^-ix)/2.

This is Euler's formula.

2) Euler's formula in the theory of complex variable functions:

e ix=cosx+isinx, e is the base of the natural logarithm, and i is the imaginary unit.

It expands the definition domain of trigonometric functions to complex numbers, establishes the relationship between trigonometric functions and exponential functions, and occupies a very important position in the complex function theory.

Replace x with -x in the formula to get :

e -ix=cosx-isinx, and then use the method of addition and subtraction of the two formulas 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+isinx as and you get:

e^i∏+1=0.

This identity, also known as Euler's formula, is one of the most fascinating formulas in mathematics that connects the most important mathematics in mathematics: two transcendental numbers: the base e of the natural logarithm, pi, and two units

The imaginary number unit i and the natural number unit 1, as well as the common 0 in mathematics. Mathematicians describe it as "God's created formula" that we can only look at but not understand.

From Euler's formula e (ix) = cosx + isinx (e is the base of the natural logarithm and i is the imaginary unit) we get:

e^(πi)=cosπ+isinπ=-1。

e ix=cosx+isinx:

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

cos x=1-x^2/2!+x^4/4!-x^6/6!……sin x=x-x^3/3!+x^5/5!-x^7/7!……Min Pao Ling.

In the formula e x, replace x with ix, so e ix=cosx isinx.

From Euler's formula e (ix) = cosx + isinx (e is the base of the natural logarithm and i is the imaginary unit) we get:

e^(πi)=cosπ+isinπ=-1。

e ix=cosx+isinx:

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

cos x=1-x^2/2!+x^4/4!-x^6/6!……sin x=x-x^3/3!+x^5/5!-x^7/7!……Min Pao Ling.

In the formula e x, replace x with ix, so e ix=cosx isinx.

Pushed with Taylor polynomials.

e ix=cosx+isinx:

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

cos x=1-x 2 Destroyer 2!+x^4/4!-x^6/6!

sin x=x-x^3/3!+x^5/5!-…In the formula e x, the change of x to the annihilation is called ix

i)^2=-1,(±i)^3=〒i,(±i)^4=1 ……Note: where " " means "subtract and add").

e^±ix=1±x/1!-x^2/2!+x^3/3!〒x^4/4!……

1-x^2/2!+…i(x-x^3/3!……The remainder is E ix=Cosx isinx