Introduced by Leonhard Euler, a Swiss mathematician, Euler’s formula can be defined by two different ways:
- For complex analysis: Euler’s formula is used in complex analysis to find the solution of complex exponential functions. It can also be termed as Euler’s identity and establishes a relationship between trigonometric functions and exponential functions.
- For Polyhedra: For a non self-intersecting polyhedra, the Euler’s formula establishes a relation between the number of faces, vertices, and edges of the polyhedra.
In this maths article, we shall learn about Euler’s formula for complex analysis and polyhedra in detail. We shall also derive Euler’s formula using different methods. Also, we will check the applications of Euler’s formula along with solved examples for better understanding of the concept.
What is Euler’s Formula?
Euler’s formula is a theorem which states that:
\(e^{ix}=\cos \theta +i\sin \theta \)
Here, ‘x’ is a real number.
‘e’ is the base of the natural log.
And ‘i’ is the imaginary number.
Euler’s formula sets up a fundamental relationship between trigonometric functions and exponential functions.
Or,
It can also be considered as a way to bridge the two representations of the same unit complex numbers in a complex plane.
Derivative of Euler’s Formula
We can derive Euler’s formula in three different ways. The first is to drive the formula using the power series. In this type of derivative exponential, sine and cosine functions are expanded as power series.
In the second tupe, we derive Euler’s formula using Calculus. Here, both the sides of the equation are treated as functions and are then differentiated.
And finally, the third derivation uses polar coordinates on the complex plane to find the values of r and \(\theta\).
Derivation 1: Using Power Series
To begin with, let us assume that power series expansion of sin z, cos z, and \(e^z\) are absolutely convergent everywhere.
For a complex variable ‘z’, the power series expansion of \(e^z\) is
\(e^z=1+\frac{z}{1!}+\frac{z^2}{2!}+\frac{z^3}{3!}+\frac{z^4}{4!}+…\)
Let us take ‘z’ as ‘ix’ (here x is any complex number).
As ‘z’ gets raised to increasing power, ‘i’ also gets raised to increasing power. Some of the powers of ‘i’ are:
\(i^0=1, i^1=i, i^2= -1, i^3= i.i^2 = -i, i^4=i^2.i^2 = 1,i^5= i.i^4=i, i^6=i.i^5 = i^2 = -1, i^7 = i.i^6 = -i\)
With z = ix the expansion \(e^z\) becomes:
\(e^{ix}=1+\frac{ix}{1!}+\frac{\left(ix\right)^2}{2!}+\frac{\left(ix\right)^3}{3!}+\frac{\left(ix\right)^4}{4!}+…\)
Using the powers of ‘i’, we get:
\(e^{ix}=1+ix-\frac{x^2}{2!}-\frac{ix^3}{3!}+\frac{x^4}{4!}+\frac{ix^5}{5!}-\frac{x^6}{6!}-\frac{ix^7}{7!}+\frac{x^8}{8!}+…\)
As the power series of expansion of \(e^z\) is absolutely convergent, let us rearrange its terms without changing the values. Let us group real and imaginary terms:
\(e^{ix}=\left(1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\frac{x^8}{8!}+…\right)+i\left(x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+…\right)\)
Let us consider power series of sine and cosine:
Power series of cos x is:
Cos x = \(1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\frac{x^8}{8!}+…\)
And the power series of sin x is:
Sin x = \(x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+…\)
Or, we can write:
\(e^{ix}=\cos x+i\sin x/latex]
Which is the statement for Euler’s formula.
Derivation 2: Using Calculus
Let us consider \(\)e^{ix}\) and cos x + i sin x as functions of x, and then differentiate them to find the common property between them.
But before that let us assume that \(e^z\), cos x, and sin x are defined and differentiable for real numbers x and complex numbers z.
Let \(f_1(x) = e^{ix}\) and \(f_2(x) = cos x +i sin x\)
Using chain rule:
\(f_1^{\prime}\left(x\right)=ie^{ix}=if_1\left(x\right)/latex] \(\)f_2^{\prime}\left(x\right)=-\sin x+i\cos x=if_2\left(x\right)\)
We can say that \(f_1(x)\) and \(f_2(x)\) satisfy f’(x) = if(x).
Consider \(\frac{f_1\left(x\right)}{f_2\left(x\right)}\) which is well defined for all x.
Using quotient rule:
\(\left(\frac{f_1}{f_2}\right)^{\prime}\left(x\right)=\frac{f_1^{\prime}\left(x\right)f_2\left(x\right)-f_1\left(x\right)f_2^{\prime}\left(x\right)}{\left[f_2\left(x\right)\right]^2}\)
\(=\frac{if_1\left(x\right)f_2\left(x\right)-f_1\left(x\right)if_2\left(x\right)}{\left[f_2\left(x\right)\right]^2}\) = 0
As derivative of \(\left(\frac{f_1}{f_2}\right)\left(x\right) = 0\), that implies \(\frac{f_1}{f_2}\) must be constant.
To find this constant, let us start with x = 0,
\(\left(\frac{f_1}{f_2}\right)\left(0\right)=\frac{e^{i0}}{\cos 0+i\sin 0}=1\)
Or,
\(\left(\frac{f_1}{f_2}\right)\left(x\right) = \frac{e^{ix}}{\cos x+i\sin x}=1\)
Moving cos x + i sin x to right, we get:
\(e^{ix}=\cos x+i\sin x\)
Hence, we get the statement of Euler’s formula.
Derivation 3: Using Polar Coordinates
To prove Euler’s formula using polar coordinates we need to treat exponentials as complex numbers under polar coordinates.
We know that all the non-zero complex numbers can be uniquely expressed in polar coordinates.
Any number \(e^{ix}\) (where x belongs to R) can be expressed as:
\(e^{ix}=r\left(\cos \theta +i\sin \theta \right)\)
Here, \(\theta\) is the principal angle from the positive real axis, and r is the radius. (Also, we know that, x = 0, then LHS = 1, r and \(\theta\) satisfy initial conditions of r(0) = 1 and \(\theta(0) = 0\).
Let us differentiate both sides of the equation:
\(ie^{ix}=\frac{dr}{dx}\left(\cos \theta +i\sin \theta \right)+r\left(-\sin \theta +i\cos \theta \right)\frac{d\theta }{dx}\)
Replace \(e^{ix}\) by \(r\left(\cos \theta +i\sin \theta \right)\) to get an expression in ‘r’ and \(\theta\).
\(ir\left(\cos \theta +i\sin \theta \right)=\left(\cos \theta +i\sin \theta \right)\frac{dr}{dx}+r\left(-\sin \theta +i\cos \theta \right)\frac{d\theta }{dx}\)
\(r\left(i\cos \theta -\sin \theta \right)=\left(\cos \theta +i\sin \theta \right)\frac{dr}{dx}+r\left(-\sin \theta +i\cos \theta \right)\frac{d\theta }{dx}\)
Equating imaginary and real parts:
\(r\cos \theta = \sin \theta \frac{dr}{dx} + r\cos \theta \frac{d\theta }{dx}\)
\(-r\sin \theta = \cos \theta \frac{dr}{dx} – r\sin \theta \frac{d\theta }{dx}\)
Let, \(\frac{dr}{dx} = \alpha\), and \(\frac{d\theta }{dx}=\beta\)
Therefore,
\(r\cos \theta = (\sin \theta) \alpha + (r\cos \theta) \beta\)…(1)
\(-r\sin \theta = (\cos \theta) \alpha – (r\sin \theta) \beta\)…(2)
Multiply (1) by \(cos\theta\) and (2) by \(sin\theta \)
\(r\cos ^2\theta =(\sin \theta \cos \theta )\alpha +(r\cos ^2\theta )\beta \)…(3)
\(-r\sin ^2\theta =(\sin \theta \cos \theta )\alpha -(r\sin ^2\theta )\beta \)…(4)
Eliminate \(\theta \) by (3) – (4):
\(r\left(\sin ^2\theta +\cos ^2\theta \right)=r\left(\sin ^2\theta +\cos ^2\theta \right)\beta \)
As \(\sin ^2\theta +\cos ^2\theta = 1\)
Therefore, \(r=r\beta /latex]
As r > 0 for all ‘x’ that means \(\)\beta = 1 or, \(\)\frac{d\theta }{dx}=1\)
Again substitute the value of \(\beta \) in (1)and (2) and compare:
\(0=\left(\sin \theta \right)\alpha \)
\(0=\left(\cos \theta \right)\alpha \)
That implies \(\alpha = 0\) or \(\frac{dr}{dx}=0\)
As \(\frac{dr}{dx}=0\), that implies ‘r’ is a constant and as \(\frac{d\theta }{dx}=1\), \(\theta =x+C\), where C is constant.
As r satisfies the condition \(r(\theta) = 1\), therefore r = 1.
And \(\theta\) satisfies the condition \(\theta(0) = 0\), so, C = 0, i.e. \(\theta = x\)
As ‘r’ and \(\theta\) are defined, therefore our original equation becomes:
\(e^{ix}=r\left(\cos \theta +i\sin \theta \right)\) = cos x + sin x
Hence, we get the statement for Euler’s formula.
Euler’s Formula for Complex Analysis
In complex analysis Euler’s formula is used to set up a relationship between trigonometric functions and exponential functions.
For any real number ‘x’, Euler’s formula is defined as:
\(e^{ix}=\cos \theta +i\sin \theta /latex]
Here, cos x, and sin x are trigonometric functions, ‘i’ is an imaginary number, and ‘e’ is the base of a natural log.
In the complex plane, the formula can be interpreted as a unit complex function \(\)e^{i\theta}\). Here \(\theta\) is a real number and is measured in radians.
Proof:
Let us prove the above result using the expansion series of \(e^x\).
\(e^x=1+\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+…\infty \)
Let us suppose that the expansion is true even if the value of ‘x’ is non-real.
Also, let \(x = i\theta\)
\(e^{i\theta }=1+\frac{i\theta }{1!}+\frac{\left(i\theta \right)^2}{2!}+\frac{\left(i\theta \right)^3}{3!}+\frac{\left(i\theta \right)^4}{4!}+…\infty \)
\(e^{i\theta }=1+\frac{i\theta }{1!}-\frac{\theta ^2}{2!}-\frac{i\theta ^3}{3!}+\frac{\theta ^4}{4!}+…\infty \) (as \( i^2 = -1\))
\(e^{i\theta }=\left(1-\frac{\theta ^2}{2!}+\frac{\theta ^4}{4!}+…\infty \right)+i\left(\frac{\theta }{1!}-\frac{\theta ^3}{3!}+\frac{\theta ^5}{5!}…\infty \right)\)
The above two series are Taylor’s expansion series for \(cos\theta \) and \(sin\theta \).
Therefore;
\(e^{ix}=\cos x+i\sin x\)
Euler’s Identity
We know that by Euler’s Formula \(e^{ix}=\cos x+i\sin x\). The formula can be expressed as identity, where \( is replaced by \(\)\pi \).
\(e^{i\pi}=\cos \pi+i\sin \pi\)
\(e^{i\pi}=-1 + i(0) \) (as \(cos \pi = -1 and sin\pi = 0\))
\(e^{i\pi} = -1\)
Or,
\(e^{i\pi} + 1=0\)
Euler’s Formula for Polyhedra
Geometrically, a polyhedron is a three-dimensional shape that has flat surfaces and straight edges. Some of the examples of polyhedra are cube, prism, cuboid, and pyramid.
The faces, edges and vertices of a polyhedron that do not self intersect are related in a certain manner.
According to Euler’s formula for polyhedra, the number of faces and vertices of a polyhedron taken together are two more than the number of edges. We can write this as:
F + V – E =2
Here, F is the number of Faces
V is the number of vertices
And E is the number of edges of a polyhedron.
Proof:
A graph is formed by drawing dots and lines. A planar graph is obtained when no lines or edges cross each other. Using the same statement a cube can be plotted as a planar graph by projecting vertices and edges on a plane. Euler’s formula graph theory states that number of dots – number of lines + number of regions cut by plane = 2.
Verification of Euler’s Formula
We can use Euler’s formula for different solid shapes and complex polyhedra.
Let us take an example of a simple polyhedron like square pyramid and triangular prism to verify Euler’s formula.
Example 1: Square Pyramid:
Number of faces = 5
Number of vertices = 5
Number of edges = 8
Therefore; F + V – E = 5 + 5 – 8 = 2
So, the Euler’s formula holds true.
Example 2: Triangular Prism
Number of faces = 5
Number of vertices = 6
Number of edges = 9
Therefore; F + V – E = 5 + 6 – 9 = 2
So, the Euler’s formula holds true.
Applications of Euler’s Formula
As Euler’s formula is an important mathematical equation, it has some interesting applications. These are:
- It is used in Euler’s identity.
- For the exponential form of complex numbers.
- As an alternative definition of trigonometric and hyperbolic functions.
- For generalization of exponential and logarithmic functions to complex numbers.
- As an alternative proof of de Moivre’s theorem and trigonometric additive identities.
Euler’s Formula Explanation
In all there are five platonic solids for which we can use Euler’s formula. These solids are:
Cube, Tetrahedron, Octahedron, Dodecahedron, and Icosahedron
The following table verifies the Euler’s formula for complex polyhedron:
SOLID | F | V | E | Euler’s Formula F + V – E |
Tetrahedron | 4 | 4 | 6 | 2 |
Cube | 6 | 8 | 12 | 2 |
Octahedron | 8 | 6 | 12 | 2 |
Dodecahedron | 12 | 20 | 30 | 2 |
Icosahedron | 20 | 12 | 30 | 2 |
Euler’s Formula Solved Examples
Que 1: A polyhedron has 12 vertices and 30 edges. Find the number of faces of the polyhedron.
Ans 1: Given that:
Number of vertices of a polyhedron = 12
Number of edges of a polyhedron = 30
We know that:
By Euler’s formula;
F + V – E = 2
Therefore,
F + 12 – 30 = 2
F = 2 – 12 + 30 = 20
Therefore, the faces of the given polyhedron are 20.
Que 2: Express \(3e^{5i}\) in (a + ib) using Euler’s formula.
Ans 2: Given that:
\(\theta = 5\)
Using Euler’s formula:
\(e^{i\theta }=\cos \theta +i\sin \theta \)
\(e^{5i} = cos 5 + i sin 5\) = 0.284 + i(-0.959) = 0.284 – 0.959i
So,
\(3e^{5i}= 0.852 – 2.877i
Therefore, we can express \(\)3e^{5i}\) in (a + ib) as 0.852 – 2.77i.
Que 3: Express \(e^{i\left(\frac{\pi }{2}\right)}\) in (a + ib) using Euler’s formula.
Ans 3: Given that:
\(\theta = \frac{\pi}{2}\)
Using Euler’s formula:
\(e^{i\theta }=\cos \theta +i\sin \theta \)
\(e^{i\left(\frac{\pi }{2}\right)} = cos(\frac{\pi}{2})+ i sin (\frac{\pi}{2})\)= 0 + \(i\times 1\) = i
Therefore, we can express \(e^{i\left(\frac{\pi }{2}\right)}\) in (a + ib) as i.
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Euler’s Formula FAQs
Q.1What’s so special about Euler’s identity?
Ans.1 Euler’s identity is considered as mathematical beauty as it makes use of five constants of maths and three maths operations and all of these occurring only once.
Q.2What is Euler’s Formula for Polyhedra?
Ans.2 According to Euler’s formula for polyhedra, the number of faces and vertices of a polyhedron taken together are two more than the number of edges. We can write this as:F + V – E =2
Here, F is the number of Faces
V is the number of vertices
And E is the number of edges of a polyhedron.
Q.3What is Euler’s Formula for the Cube?
Ans.3 Cube is an example of platonic solids, so Euler’s formula for a cube is similar to any other polyhedron, i.e F + V – E = 2.
Q.4What Does Euler’s Formula Mean?
Ans.4 Euler’s formula is a theorem which states that:\(e^{ix}=\cos \theta +i\sin \theta \)
Here, ‘x’ is a real number.
‘e’ is the base of the natural log.
And ‘i’ is the imaginary number.
Euler’s formula sets up a fundamental relationship between trigonometric functions and exponential functions.
Q.5What is Euler’s Identity?
Ans.5 We know that by Euler’s Formula \(e^{ix}=\cos x+i\sin x\). The formula can be expressed as identity, where \( is replaced by \(\)\pi \).
\(e^{i\pi}=\cos \pi+i\sin \pi\)
\(e^{i\pi}=-1 + i(0) \) (as \(cos \pi = -1 and sin\pi = 0\))
\(e^{i\pi} = -1\)
Or,
\(e^{i\pi} + 1=0\)
FAQs
What is the Euler's formula for a polyhedron? ›
This theorem involves Euler's polyhedral formula (sometimes called Euler's formula). Today we would state this result as: The number of vertices V, faces F, and edges E in a convex 3-dimensional polyhedron, satisfy V + F - E = 2.
What is Euler's formula explain with examples? ›It is written F + V = E + 2, where F is the number of faces, V the number of vertices, and E the number of edges. A cube, for example, has 6 faces, 8 vertices, and 12 edges and satisfies this formula.
Why there is always 2 answers to Euler's polyhedral formula? ›There's a relationship between the number of vertices, faces, and edges on a polyhedron* that will always result in the number '2'. Knowing the relationship results in a formula.
What is Euler formula in complex analysis? ›Euler's law states that 'For any real number x, e^ix = cos x + i sin x. This complex exponential function is sometimes denoted cis x ("cosine plus i sine"). The formula is still valid if x is a complex number.
What is the formula for Euler's method? ›y=y(xi)+f(xi,y(xi))(x−xi).
What is the Euler formula for polynomials? ›The Euler polynomials En(x) and the Euler numbers En are defined by the following generating functions:(40)2exzez+1=∑n=0∞En(x)znn!
What is the answer to Euler's identity? ›Euler's formula is eⁱˣ=cos(x)+i⋅sin(x), and Euler's Identity is e^(iπ)+1=0.
Is Euler's formula true for all polyhedron? ›Given, Euler's formula is true for all three-dimensional shapes. E for number of edges. Euler's formula is true for the cube and the icosahedron. Therefore, Euler's formula is true only for polyhedra.
What are the three parts of a polyhedron that are related by Euler's formula? ›There is a relationship between the number of faces, edges, and vertices in a polyhedron, which can be presented by a math formula known as “Euler's Formula.”
How do you find the number of faces in a polyhedron? ›Use the fact that V−E+F=2 V − E + F = 2 V-E+F=2 V−E+F=2 to calculate the number of faces for the polyhedron where V V V V is the number of vertices, E E E E is the number of edges and F F F F is the number of faces of the polyhedron.
How do you write a complex number in Euler? ›
Euler's Formula states that eiφ=cosφ+isinφ for any real number φ. This formula is one of the most important contributions to complex analysis − and it will be very helpful when you are trying to solve equations with complex numbers!
Why do we use Euler's formula? ›Euler's formula relates the complex exponential to the cosine and sine functions. This formula is the most important tool in AC analysis. It is why electrical engineers need to understand complex numbers. Created by Willy McAllister.
How do you remember Euler's formula? ›- 2.7 1828 1828. And following THAT are the digits of the angles 45°, 90°, 45° in a Right-Angled Isosceles Triangle (no real reason, just how it is):
- 2.7 1828 1828 45 90 45. (An instant way to seem really smart!)
We can also find the number of faces in a polyhedron using Euler's formula, F + V - E = 2, if we know the number of edges and vertices.
Which of the following is a polyhedron answer? ›Cuboid is a polyhedron because its faces are congruent and regular polygons.
What is a polyhedron with 4 faces? ›The smallest polyhedron is the tetrahedron with 4 triangular faces, 6 edges, and 4 vertices.
What is a polyhedron with 12 faces? ›In geometry, a dodecahedron (Greek δωδεκάεδρον, from δώδεκα dōdeka "twelve" + ἕδρα hédra "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid.
What is a polyhedron with 5 faces? ›A pentahedron is polyhedron having five faces. Because there are two pentahedral graphs, there are two convex pentahedra, corresponding to the topologies of the square pyramid and the triangular prism.
How do you solve complex numbers examples? ›Complex Numbers in Maths. Complex numbers are the numbers that are expressed in the form of a+ib where, a,b are real numbers and 'i' is an imaginary number called “iota”. The value of i = (√-1). For example, 2+3i is a complex number, where 2 is a real number (Re) and 3i is an imaginary number (Im).
What is an example of Euler's identity? ›For x = π 2 , we have e i π 2 = cos π 2 + i sin π 2 = i . This result is useful in some calculations related to physics. For , we have e i π = cos π + i sin , which means that e i π = − 1 . This result is equivalent to the famous Euler's identity.
What is the formula for complex numbers? ›
The standard form of writing a complex number is z = a + ib. The standard form of the complex number has two parts, the real part, and the imaginary part. In the complex number z = a + ib, a is the real part and ib is the imaginary part.
What is complex analysis simple explanation? ›Complex analysis is the study of complex numbers together with their derivatives, manipulation, and other properties. Complex analysis is an extremely powerful tool with an unexpectedly large number of practical applications to the solution of physical problems.
Which is the analysis method for complex? ›pH titration method-
of complex is indicated by change in pH of the mixture.
Complex analysis is used to solve the CPT Theory (Charge, Parity and Time Reversal), as well as in conformal field theory and in the Wick's Theorem. Complex variables are also a fundamental part of QM as they appear in the Wave Equation. String Theory: Indeed, Complex Analysis shows up in abundance in String theory.
What does Euler's method tell us? ›Euler's method is used for approximating solutions to certain differential equations and works by approximating a solution curve with line segments. In the image to the right, the blue circle is being approximated by the red line segments.
How to calculate eulers number? ›Euler's Number (e)
The Euler's number 'e', is the limit of (1 + 1/n)^{n} as n approaches infinity, an expression that arises in the study of compound interest. It can also be expressed as the sum of infinite numbers. e = ∑ n = 0 ∞ 1 n ! = 1 1 + 1 1 + 1 1.2 + 1 1.2 .
The Euler equations can be applied to incompressible or compressible flow. The incompressible Euler equations consist of Cauchy equations for conservation of mass and balance of momentum, together with the incompressibility condition that the flow velocity is a solenoidal field.
What is the formula for any polyhedron? ›He found that υ − e + f = 2 for every convex polyhedron, where υ, e, and f are the numbers of vertices, edges, and faces of the polyhedron.
Is Euler's formula always 2? ›It doesn't always add to 2. The reason it didn't work was that this new shape is basically different ... that joined bit in the middle means that two vertices become 1.
What is Euler's rule for 3d shapes? ›According to Euler's formula for any convex polyhedron, the number of Faces (F) and vertices (V) added together is exactly two more than the number of edges (E).
What is Euler's number simplified? ›
An irrational number represented by the letter e, Euler's number is 2.71828..., where the digits go on forever in a series that never ends or repeats (similar to pi). Euler's number is used in everything from explaining exponential growth to radioactive decay.
What is a polyhedron example? ›A three-dimensional shape with flat polygonal faces, straight edges, and sharp corners or vertices is called a polyhedron. Common examples are cubes, prisms, pyramids. However, cones, and spheres are not polyhedrons since they do not have polygonal faces.
Does Euler's formula work for all polyhedron? ›Euler's formula is true for the cube and the icosahedron. It turns out, rather beautifully, that it is true for pretty much every polyhedron. The only polyhedra for which it doesn't work are those that have holes running through them like the one shown in the figure below.
Why is Euler's formula so important? ›Euler's formula establishes the fundamental relationship between trigonometric functions and exponential functions. Geometrically, it can be thought of as a way of bridging two representations of the same unit complex number in the complex plane.
How do you find the value of e? ›The number e is approximately 2.71828, and is the base of natural logarithms. It is also one of the most important numbers in mathematics. The value of e can be found when taking the so-called "limit definition". The value of e has many applications in calculus, physics, and engineering.