a^r/(a-b)(a-c)+b^r/(b-c)(b-a)+c^r/(c-a)(c-b)
When r=0, 1, the value of the formula is 0; When r=2, the value is 1.
When r=3, the value is a+b+CB+C.
E ix = cosx+isinx, e is the base of natural logarithm, and I is the imaginary unit. It extends the definition domain of trigonometric function to complex number, and establishes the relationship between trigonometric function and exponential function, which occupies a very important position in the theory of complex variable function.
Proof of e IX = cosx+isinx;
Because e x = 1+x/ 1! +x^2/2！ +x^3/3！ +x^4/4！ +……
Because 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, change x into IX. (I) 2 =- 1, (I) 3 = 〒 I, (I) 4 = 1 ... (Note: "〒" means ".
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 add and subtract two formulas to get: SINX = (E IX-E IX)/(2i), COSX = (E IX+E IX)/2. These two formulas are also called Euler formula. Let x in e ix = cosx+isinx be ∏, and you get:
E I π+ 1 = 0。 This identity is also called Euler formula.
Euler formula in triangle
Let r be the radius of the circumscribed circle of the triangle, r be the radius of the inscribed circle, and d be the distance from the outer center to the inner center, then: D 2 = R 2-2rr.
Euler formula in topology
V+F-E=X(P), v is the number of vertices of polyhedron p, f is the number of faces of polyhedron p, e is the number of sides of polyhedron p, and X(P) is the Euler characteristic of polyhedron p.
If P can be homeomorphism on a sphere (which can be understood as expansion and stretching on a sphere), then X (P) = 2; If p is homeomorphic on a sphere with h ring handles, then X(P)=2-2h.
X(P), called Euler characteristic of P, is a topological invariant, that is, a quantity that will not change no matter how topological deformation is carried out, which is the scope of topological research.
Application in polyhedron:
There is a relationship between the number of vertices v, the number of faces f and the number of edges e of a simple polyhedron.
V+F-E=2
This formula is called Euler formula.
Euler formula in elementary number theory
Euler φ function: φ(n) is the integer number of n coprime in all positive integers less than n, and n is a positive integer.
Euler proved the following formula:
If the factorization of the standard prime factor of n is p1a1* p2a2 * ... * pmam, all PJ (j = 1, 2, ..., m) are prime numbers, and pairwise is not equal. Then there is
φ(n)= n( 1- 1/p 1)( 1- 1/p2)……( 1- 1/pm)
It can be proved by the principle of inclusion and exclusion.
In addition, many famous theorems are named after Euler.
(6) Euler formula in three-dimensional graphics:
Number of faces+number of vertices -2 = number of edges.