The process of deriving Euler's formula

Derivation of Euler’s formula

e^ix=cosx isinx, where e is the base of natural logarithms and i is the imaginary unit. This equation extends the domain of trigonometric functions to complex numbers and establishes the relationship between trigonometric and exponential functions. In the theory of functions of complex variables, this equation plays an important role.

e^ix=Proof of 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! -……

In the expansion of e^x, replace x with ±ix. (±i)^2=-1, (±i)^3=〒i, (±i)^4=1... (Note : Where "〒" means "subtract plus")

e^±ix=1±x/1!-x^2/2! x^3/3! 〒x^4/4! ……

=(1-x^2/2!…)±i(x-x^3/3!…)

So e^±ix=cosx±isinx

Replace x in the formula with -x to get:

e^-ix=cosx-isinx, and then use the method of addition and subtraction of the two equations to get: sinx=(e^ix-e^-ix)/(2i), cosx=(e^ix e^-ix )/2. These two are also called Euler's formulas. Taking x in e^ix=cosx isinx as ∏ we get:

e^iπ 1=0.

Algorithm: Euler Road

Eulerian circuit [Definition]

A loop in graph G, if it passes through each edge in G exactly once, is called an Euler loop.

The graph with Euler circuit is called Euler graph (referred to as E graph).

【Related conclusions】

theorem:

An undirected graph is an Euler graph if and only if the degree of all vertices of the graph is even.

A directed graph is an Euler graph if and only if the degree of all vertices of the graph is 0.

A solution to the Euler circuit

The following is the Euler loop output code of the undirected graph: Note that the premise of the output is that the graph has been judged to be a Euler loop.

int num = 0; //Mark output queue

int match[MAX];//The degree of the marked node, undirected graph, does not distinguish between in-degree and out-degree

void solve(int x)

l{

l if(match[x] == 0)

l

l Record[num] = x;

l

l else

l {

l for(int k =0;kl {

l if(Array[x][k] !=0 )

l {

l Array[x][k]--;

l Array[k][x]--;

l match[x]--;

l match[k]--;

l solve(k);

l }

l

l }

l Record[num] = x;

l }

l}

Note that the points in the record are arranged in order of output. Therefore, if you want to output the Euler path, you need to output the record upside down.

The idea of ​​Euler circuit:

Find the starting point in a loop. Start from a certain node, and then find a loop path from this point back to this point. This method ensures that every edge is traversed. If there is an edge at a certain point that has not been traversed, let this point be the starting point, this edge be the starting edge, and connect it to the current ring. This continues until all edges have been traversed. In this way, the entire graph is connected together.

Specific steps:

1. If there is no point connected to this point at this time, then add it to the path

2. If the point has connected points, then make a list and traverse these points until there are no connected points.

3. Process the current point, delete the edge traveled, perform the same operation on its adjacent points, and add the deleted points to the path.

4. This is actually a recursive process.

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