# Eulers Formula

## Euler’s Formula

Eulers Formula, named after Leonhard Euler, is a mathematical formula in complex analysis that provides the essential link between trigonometric functions and complex exponential functions. The essential link between trigonometric functions and complex exponential functions is established by Euler’s formula. One of the most fundamental equations in Mathematics and Engineering, Eulers Formula has a wide range of uses. A Swiss mathematician named Leonhard Euler provided the Eulers Formula. Nevertheless, there are two versions of Eulers Formula, one for polyhedrons and the other for complex analysis. It is a crucial formula for solving complex exponential functions in complex analysis. Another name for Eulers Formula is Euler’s identity. It is employed to establish the connection between complex exponential functions and trigonometric functions. For polyhedra, Eulers Formula states that the number of faces, vertices, and edges is connected in a certain way for every polyhedron that does not self-intersect.

Roger Cotes, an English Mathematician, made a geometrical argument in 1714 that can be translated as follows:

ix = ln (cos x + I sin x).

This equation’s exponentiation results in Eulers Formula. Due to the fact that a complex logarithm might have an unlimited number of values that differ by multiples of 2i, it should be noted that the logarithmic statement is not always true for complex numbers. By contrasting the series expansions of the exponential and trigonometric formulas, Leonhard Euler, who began studying the exponential function around 1740, arrived at the equation that bears his name. His seminal book ‘Introductio in analysin infinitorum’, which served as the formula’s first publication, was written in 1748. The correlation between Bernoulli and Euler, who was familiar with the aforementioned equation, demonstrates Bernoulli’s lack of proficiency with complex logarithms. Complex logarithms, according to Euler, can have an unlimited number of values. Caspar Wessel described the concept of complex numbers as points in the complex plane roughly 50 years later.

Some authors refer to the more general complex version of the formula as Eulers Formula since it is still true even if x is a complex number. Eulers Formula is widely used in Mathematics, Physics, and Engineering. The equation was dubbed “our treasure” and “the most astounding formula in mathematics” by the physicist Richard Feynman.

## Euler’s Formula Equation

Here are two possible Eulers Formula that are applied in various situations.

1. i)   eix = cos x + i sin x, which is Eulers Formula for complex analysis.
2. ii)  faces + vertices – edges = 2, which is Eulers Formula for polyhedra.

According to Euler’s identity or formula, the complex analysis result for any real number x is represented by:

eix = cos x + I sin x

where,

x is an actual number.

e is the natural logarithm’s base.

sin x and cos x are trigonometric functions.

I is an imaginary unit

The term “cis x” is frequently used to refer to the formula cos x + I sin x.

According to Euler’s polyhedral formula, there are precisely two more faces and vertices combined than there are edges. The polyhedron’s Eulers Formula is as follows:

Edges + 2 = Faces + Vertices

F + V = E + 2

Or

F + V – E = 2

where,

F stands for “faces”

V is the number of vertices.

E is the number of edges.

### What is Euler’s Formula?

The essential connection between trigonometric and exponential functions is established by Eulers Formula. It can be conceptualised geometrically as a means of connecting two complex plane representations of the same unit complex number. When integrating trigonometric expressions in the context of complex numbers, one is likely to run into the so-called Eulers Formula. This remarkable equation, which bears the name of the eminent Mathematician Leonhard Euler, merits a deeper look so as to get the most out of it. Eulers Formula enables students to represent complex numbers as exponentials and investigate the various methods by which it can be fairly simply established. Eulers Formula can be understood as two distinct ways of running around the unit circle to get to the same point since increasing a unit complex number to power can be thought of as repeated multiplications (in this example, adding up angles).

There are a minimum of three approaches to establishing the Eulers Formula. The first derivation uses power series to show that the formula is valid by expanding the exponential, sine, and cosine functions as power series. Calculus is used in the second derivation of Eulers Formula so that both sides of the problem can be considered as functions and differentiated appropriately. The discovery of a shared characteristic follows, which may be used to demonstrate that both functions are actually equal. Polar coordinates in the complex plane are used in yet another derivation of Eulers Formula, which leads to the discovery of the values of r. In fact, by simply looking at the formula, one might be able to infer what these values are.

### Euler’s Formula For complex analysis

The Eulers Formula of a complex number is significant enough in Mathematics. It is a very practical model that allows many calculations to be simplified. The link between complex exponential functions and trigonometric functions is established using Euler’s formula in complex analysis. For each real number x, Eulers Formula is defined and can be expressed as:

eix = cos x + I sin x

Here, I is the imaginary unit, cos and sin are trigonometric functions, and e is the natural logarithm’s base.

This equation can be understood as a unit complex function tracing a unit circle in a complex plane, where is a real value expressed in radians. This illustration may at first be unclear. Raising a real number to an imaginary number makes no sense. One can, however, take comfort in the fact that there is a solid rationale for this relationship. One can use rough proof to demonstrate why it should be true.

### Euler’s Formula for Polyhedrons

The three-dimensional solid objects known as polyhedra have smooth surfaces and straight edges. Examples include the cube, cuboid, prism, and pyramid. There is a certain relationship between the number of faces, vertices, and edges for every polyhedron that does not self-intersect. The number of vertices and faces combined is exactly two greater than the number of edges, according to Euler’s formula for polyhedra. One can write the Eulers Formula for a polyhedron as follows:

F + V – E = 2

Here, the number of faces, vertices, and edges are denoted by the letters F, V, and E respectively.

When one only draws dots and lines, it turns into a graph. When there are no intersections of any lines or edges, one can have a planar graph. By transferring the vertices and edges of a cube onto a plane, one can represent it as a planar graph. The number of dots minus the number of lines plus the number of regions the plane is divided into equals two, according to Euler’s formula for graph theory.

### What is Euler’s Formula Used For?

Eulers Formula, one of the most significant mathematical equations, has a number of fascinating applications in a variety of fields. Among them are the well-known Euler’s identity, exponential representation of complex numbers, and alternative trigonometric and hyperbolic function definitions. Eulers Formula is also used in complex number generalisation of exponential and logarithmic functions, De Moivre’s Theorem alternative proofs and trigonometric additive identities. The more general formula is F + V –  E = X, where X is the Euler characteristic. F + V – E can equal 2 or 1 and have other values. In order to examine any three-dimensional space, not just polyhedra, students can validate Euler’s formula. There are precisely 5 regular polyhedra, according to Euler’s graph theory. One may check to see if there is a simple polyhedron with 10 faces and 17 vertices using the Eulers Formula calculator. The prism has 10 faces and 16 vertices, with an octagon serving as its base. Even though the final answer is a real function involving sine and cosine, the formula is frequently used in differential equations to simplify answers. This is because the exponential function is an eigenfunction of the differentiation operation. Signals that change periodically over time are frequently described as a combination of sinusoidal functions in electrical engineering, signal processing, and related fields. However, Eulers Formula more conveniently expresses these as the sum of exponential functions with imaginary exponents. Additionally, Euler’s formula can be used in the phasor analysis of circuits to express the impedance of a capacitor or an inductor.

There is a sphere of imaginary units in the four-dimensional quaternion space. Euler’s formula is applicable for any point r on this sphere. The element is known as a versor in quaternions and is

exp xr=cos x + r sin x.

In the 4-space, the set of all versors creates a 3-sphere.

### Verification of Euler’s Formula for Solids

Examples of Euler’s formulas include complex polyhedra and solid forms. One can check the formula for a few basic polyhedra, like a triangle and a square pyramid.

There are 5 faces, 5 vertices, and 8 edges in a square pyramid.

F + V − E = 5 + 5 − 8 = 2

For a triangle with five faces, six vertices, and nine edges.

F + V − E = 5 + 6 − 9 = 2

### Euler’s Formula Explanation

It is possible to demonstrate Eulers Formula for 5 platonic solids. They are the icosahedron, cube, tetrahedron, octahedron, and octahedron. It is possible to test Eulers Formula using these complex polyhedra that serve as examples. The utility problem is used to demonstrate Eulers Formula: The three utilities—gas (G), water (W), and electricity (E)—are to be linked to the three houses (H1, H2, and H3). They must be linked together so that no pipe crosses the other pipe. One removes an edge to form a tree in order to obtain a full cycle in any planar graph with no intersection. Edges and faces are both reduced by one as a result, leaving vertices – edges + faces = constant. This procedure is repeated until a tree is all that is left of the original graph. The Euler characteristic, or vertices –  edges + faces = 2, is what we get at the end. Eulers Formula for graph theory should be used with the utility graph.

Power series are used in one of the most simple derivations of the Eulers Formula. It entails increasing the exponential, sine, and cosine power series in order to arrive at the conclusion that equality is true. This method comes with a warning because it presumes that the power series expansions are perfectly convergent everywhere (e.g., that they hold for all complex numbers). It also has the benefit of demonstrating that Euler’s formula is valid for all complex numbers. Another interesting method for establishing Euler’s formula is to consider both LHS and RHS of the Eulers Formula as functions of x, before differentiating them in order to uncover some common properties between them. However, one must first suppose that the functions are defined and differentiable for all real numbers and complex numbers in order for that to happen. It is also easy to demonstrate that Euler’s formula holds for all complex numbers by presuming that these functions are differentiable for all complex numbers. Treating exponentials as numbers, or more precisely as complex numbers under polar coordinates, is yet another clever way to demonstrate Euler’s formula. In fact, students already know that there exists a unique technique to express any non-zero complex integers in polar coordinates.

Eulers Formula is a rare jewel in the field of Mathematics, as was demonstrated above. It lays the groundwork for significant advancement in the fields of complex numbers, complex functions, and related theories by establishing the essential link between exponential and trigonometric functions. In fact, Euler’s formula appears to cover every possible angle whenever expressions like sin i, i, and e are included, regardless of whether it is Euler’s identity or a complex logarithm. It is an effective instrument with great potential for reward, making it a legitimate contender for the title of “the most remarkable formula in Mathematics.”

### Solved Examples Using Euler’s Formula

Solved examples using the Eulers Formula are available for practice on the Extramarks website and mobile application.