Learn about Euler’s Formula

Euler’s Function

About: Leonhard Euler 

Leonhard Euler was a known Swiss mathematician, astronomer, logician, geographer, physicist and engineer who discovered and encouraged many other departments of mathematics, that included the analytic number theory, complex analysis of various numbers, and finite and infinite calculus.

What is Euler’s Formula?

The essential relationship between trigonometric functions and complex exponential functions is determined by Euler’s formula. Euler’s formula, sometimes known as Euler’s equation, is a fundamental equation in mathematics and computer science with numerous applications. There are two Euler’s formulas, one for rigorous calculations and another for polyhedron analysis.

There are two ways to “prove” Euler’s formula. In terms of known power series expansions, expand the left and right sides of Euler’s formula. Compare and contrast powers of equivalent magnitude.

Demonstrate that both sides of Euler’s formula are solutions of the same second-order linear differential equation with constant coefficients. Because there are only two linearly independent solutions to a second-order linear problem, write

Leonhard Euler stated two formulae, namely-

Euler’s Formula for Complex Analysis of Numbers.

Euler’s Formula for Polyhedrons.

Euler’s Formula for Complex Analysis of Numbers

Euler’s Formula for Complex Analysis of Numbers was a formula given by Leonhard Euler to solve complex exponent based functions. The other name for Euler’s Formula for Complex Analysis of Numbers is Euler’s Identity Formula. 

The association and kind of relationship between a Trigonometric function and complex exponent based function are found by using this formula.

Euler’s formula of a real or complex is sufficient to justify its section. It is a really useful concept that simplifies a lot of calculations and makes work easy. 

In complex analysis, Euler’s formula is used to calculate the relationship between trigonometric functions and complex exponential functions.

Any real number say for instance ‘x’ can be written as-

eix = cos x + i sin x

In the above formula cos and sin are two known trigonometric functions, ‘e’ is the base of a specific logarithm which is natural and ‘i’ is an imaginary unit.

Euler’s Formula for Polyhedrons

The number of faces, vertices and edges of a given polyhedron which is not a self-intersecting polyhedron their dimensions are related using Euler’s Formula for Polyhedrons. Another name for Euler’s Formula for Polyhedrons is Euler’s Characteristic Formula.

A polyhedron is a three-dimensional figure or a solid shape that has several flat surfaces and edges that are straight. 

The number of faces, vertices and edges of every polyhedron which do not self-intersect itself is correlated in a specific way. 

The number of vertices and faces together is exactly two greater than the number of edges, according to Euler’s formula for polyhedra.

Euler’s formula for polyhedrons can be written as-

Faces + Vertices – Edges = 2

Here, 

‘F’ denotes the number of faces in a Polyhedron.

‘E’ denotes the number of edges in a Polyhedron.

‘V’ denotes the number of vertices in a Polyhedron.

Euler’s Identity

From Euler’s Formula for Complex Analysis of Numbers, we have 

six = cos x + sin x. 

The given formula gives us an identity when x is replaced with π:

eiπ = cos π + isin π

eiπ = -1 + i (0)                                               …. (since sin π = 0 and cos π = -1)

eiπ = -1 or eiπ + 1 = 0.  This is known as Euler’s identity.

Euler’s Formula

Any complex number x may be written as sin x, which rests on a unit circle with real and imaginary components cost and sin x, respectively, credits go to Euler’s formula. Rotations along the unit circle can thus be interpreted as various operations (such as finding the roots of unity).

Applications of Euler’s Formula

An application of Euler’s Formula is that it is used to know the definition of the trigonometric based functions.

Euler’s formula also helps us to derive several trigonometric identities with ease.

In a plane drawing of a connected planar graph, the number of vertices ‘v’, number of edges ‘e’ and number of graphical regions ‘r’. In a simple connected graph with the shortest circuit length, the Euler’s Formula can be used to prove that the graph is not planar.

Conclusion 

Euler’s Formula plays an important role in the field of Mathematical Sciences. One must know about the basics and all the concepts of the given topic to understand in detail the key points of the topic.

There are two ways to “prove” Euler’s formula. In terms of known power series expansions, expand the left and right sides of Euler’s formula. Compare and contrast powers of equivalent magnitude.

Euler’s Formula does help and plays an important role in the field of mathematics and algebra.