Euler's formula

Template:Short description Template:About Template:Use dmy dates Template:E (mathematical constant) Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that, for any real number Template:Mvar, one has $$e^{ix}=\cos x+i\sin x,$$ where Template:Mvar is the base of the natural logarithm, Template:Mvar is the imaginary unit, and Template:Math and Template:Math are the trigonometric functions cosine and sine respectively. This complex exponential function is sometimes denoted Template:Math ("cosine plus i sine"). The formula is still valid if Template:Mvar is a complex number, and is also called Euler's formula in this more general case.

Euler's formula is ubiquitous in mathematics, physics, chemistry, and engineering. The physicist Richard Feynman called the equation "our jewel" and "the most remarkable formula in mathematics".

When Template:Math, Euler's formula may be rewritten as Template:Math or Template:Math, which is known as Euler's identity.

History

In 1714, the English mathematician Roger Cotes presented a geometrical argument that can be interpreted (after correcting a misplaced factor of $\sqrt{-1}$) as: $$ix=\ln (\cos x+i\sin x).$$ Exponentiating this equation yields Euler's formula. Note that the logarithmic statement is not universally correct for complex numbers, since a complex logarithm can have infinitely many values, differing by multiples of Template:Math.

Visualization of Euler's formula as a helix in three-dimensional space. The helix is formed by plotting points for various values of $\theta$ and is determined by both the cosine and sine components of the formula. One curve represents the real component ($\cos \theta$) of the formula, while another curve, rotated 90 degrees around the z-axis (due to multiplication by $i$), represents the imaginary component ($\sin \theta$).

Around 1740 Leonhard Euler turned his attention to the exponential function and derived the equation named after him by comparing the series expansions of the exponential and trigonometric expressions. The formula was first published in 1748 in his foundational work Introductio in analysin infinitorum.

Johann Bernoulli had found that $$\frac{1}{1+x^2}=\frac{1}{2}(\frac{1}{1-ix}+\frac{1}{1+ix}).$$

And since $$\int \frac{dx}{1+ax}=\frac{1}{a}\ln (1+ax)+C,$$ the above equation tells us something about complex logarithms by relating natural logarithms to imaginary (complex) numbers. Bernoulli, however, did not evaluate the integral.

Bernoulli's correspondence with Euler (who also knew the above equation) shows that Bernoulli did not fully understand complex logarithms. Euler also suggested that complex logarithms can have infinitely many values.

The view of complex numbers as points in the complex plane was described about 50 years later by Caspar Wessel.

Definitions of complex exponentiation

The exponential function Template:Math for real values of Template:Mvar may be defined in a few different equivalent ways (see Characterizations of the exponential function). Several of these methods may be directly extended to give definitions of Template:Math for complex values of Template:Mvar simply by substituting Template:Mvar in place of Template:Mvar and using the complex algebraic operations. In particular, we may use any of the three following definitions, which are equivalent. From a more advanced perspective, each of these definitions may be interpreted as giving the unique analytic continuation of Template:Math to the complex plane.

Differential equation definition

The exponential function $f(z)=e^z$ is the unique differentiable function of a complex variable for which the derivative equals the function $$\frac{df}{dz}=f$$ and $$f(0)=1.$$

Power series definition

For complex Template:Mvar $$e^z=1+\frac{z}{1!}+\frac{z^2}{2!}+\frac{z^3}{3!}+\cdots =\sum _{n=0}^{\mathrm{\infty }}\frac{z^n}{n!}.$$

Using the ratio test, it is possible to show that this power series has an infinite radius of convergence and so defines Template:Math for all complex Template:Mvar.

Limit definition

For complex Template:Mvar $$e^z=\underset{n\to \mathrm{\infty }}{\lim }{(1+\frac{z}{n})}^n.$$

Here, Template:Mvar is restricted to positive integers, so there is no question about what the power with exponent Template:Mvar means.

Proofs

Various proofs of the formula are possible.

Using differentiation

This proof shows that the quotient of the trigonometric and exponential expressions is the constant function one, so they must be equal (the exponential function is never zero, so this is permitted).

Consider the function Template:Math $$f(\theta )=\frac{\cos \theta +i\sin \theta }{e^{i\theta }}=e^{-i\theta }(\cos \theta +i\sin \theta )$$ for real Template:Mvar. Differentiating gives by the product rule $$f^{\prime }(\theta )=e^{-i\theta }(i\cos \theta -\sin \theta )-i{e}^{-i\theta }(\cos \theta +i\sin \theta )=0$$ Thus, Template:Math is a constant. Since Template:Math, then Template:Math for all real Template:Mvar, and thus $$e^{i\theta }=\cos \theta +i\sin \theta .$$

Using power series

A plot of the first few terms of the Taylor series of Template:Math for Template:Math.

Here is a proof of Euler's formula using power-series expansions, as well as basic facts about the powers of Template:Mvar: $$\begin{array}{cccccccc}{i}^0& =1,& i^1& =i,& i^2& =-1,& i^3& =-i,\\ i^4& =1,& i^5& =i,& i^6& =-1,& i^7& =-i\\ & ⋮& & ⋮& & ⋮& & ⋮\end{array}$$

Using now the power-series definition from above, we see that for real values of Template:Mvar $$\begin{array}{cc}{e}^{ix}& =1+ix+\frac{(ix{)}^2}{2!}+\frac{(ix{)}^3}{3!}+\frac{(ix{)}^4}{4!}+\frac{(ix{)}^5}{5!}+\frac{(ix{)}^6}{6!}+\frac{(ix{)}^7}{7!}+\frac{(ix{)}^8}{8!}+\cdots \\ & =1+ix-\frac{x^2}{2!}-\frac{i{x}^3}{3!}+\frac{x^4}{4!}+\frac{i{x}^5}{5!}-\frac{x^6}{6!}-\frac{i{x}^7}{7!}+\frac{x^8}{8!}+\cdots \\ & =(1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\frac{x^8}{8!}-\cdots )+i(x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\cdots )\\ & =\cos x+i\sin x,\end{array}$$ where in the last step we recognize the two terms are the Maclaurin series for Template:Math and Template:Math. The rearrangement of terms is justified because each series is absolutely convergent.

Using polar coordinates

Another proof is based on the fact that all complex numbers can be expressed in polar coordinates. Therefore, for some Template:Mvar and Template:Mvar depending on Template:Mvar, $$e^{ix}=r(\cos \theta +i\sin \theta ).$$ No assumptions are being made about Template:Mvar and Template:Mvar; they will be determined in the course of the proof. From any of the definitions of the exponential function it can be shown that the derivative of Template:Math is Template:Math. Therefore, differentiating both sides gives $$i{e}^{ix}=(\cos \theta +i\sin \theta )\frac{dr}{dx}+r(-\sin \theta +i\cos \theta )\frac{d\theta }{dx}.$$ Substituting Template:Math for Template:Math and equating real and imaginary parts in this formula gives Template:Math and Template:Math. Thus, Template:Mvar is a constant, and Template:Mvar is Template:Math for some constant Template:Mvar. The initial values Template:Math and Template:Math come from Template:Math, giving Template:Math and Template:Math. This proves the formula $$e^{i\theta }=1(\cos \theta +i\sin \theta )=\cos \theta +i\sin \theta .$$

Applications

Applications in complex number theory

Euler's formula Template:Math illustrated in the complex plane.

Interpretation of the formula

This formula can be interpreted as saying that the function Template:Math is a unit complex number, i.e., it traces out the unit circle in the complex plane as Template:Mvar ranges through the real numbers. Here Template:Mvar is the angle that a line connecting the origin with a point on the unit circle makes with the positive real axis, measured counterclockwise and in radians.

The original proof is based on the Taylor series expansions of the exponential function Template:Math (where Template:Mvar is a complex number) and of Template:Math and Template:Math for real numbers Template:Mvar (see above). In fact, the same proof shows that Euler's formula is even valid for all complex numbers Template:Mvar.

A point in the complex plane can be represented by a complex number written in cartesian coordinates. Euler's formula provides a means of conversion between cartesian coordinates and polar coordinates. The polar form simplifies the mathematics when used in multiplication or powers of complex numbers. Any complex number Template:Math, and its complex conjugate, Template:Math, can be written as $$\begin{array}{cc}z& =x+iy=|z|(\cos \phi +i\sin \phi )=r{e}^{i\phi },\\ \overline{z}& =x-iy=|z|(\cos \phi -i\sin \phi )=r{e}^{-i\phi },\end{array}$$ where

• Template:Math is the real part,
• Template:Math is the imaginary part,
• Template:Math is the magnitude of Template:Mvar and
• Template:Math.

Template:Mvar is the argument of Template:Mvar, i.e., the angle between the x axis and the vector z measured counterclockwise in radians, which is defined up to addition of Template:Math. Many texts write Template:Math instead of Template:Math, but the first equation needs adjustment when Template:Math. This is because for any real Template:Mvar and Template:Mvar, not both zero, the angles of the vectors Template:Math and Template:Math differ by Template:Pi radians, but have the identical value of Template:Math.

Use of the formula to define the logarithm of complex numbers

Now, taking this derived formula, we can use Euler's formula to define the logarithm of a complex number. To do this, we also use the definition of the logarithm (as the inverse operator of exponentiation): $$a=e^{\ln a},$$ and that $$e^a{e}^b=e^{a+b},$$ both valid for any complex numbers Template:Mvar and Template:Mvar. Therefore, one can write: $$z=\left|z\right|e^{i\phi }=e^{\ln \left|z\right|}{e}^{i\phi }=e^{\ln \left|z\right|+i\phi }$$ for any Template:Math. Taking the logarithm of both sides shows that $$\ln z=\ln \left|z\right|+i\phi ,$$ and in fact, this can be used as the definition for the complex logarithm. The logarithm of a complex number is thus a multi-valued function, because Template:Mvar is multi-valued.

Finally, the other exponential law $${\left(e^a\right)}^k=e^{ak},$$ which can be seen to hold for all integers Template:Mvar, together with Euler's formula, implies several trigonometric identities, as well as de Moivre's formula.

Relationship to trigonometry

Relationship between sine, cosine and exponential function

Euler's formula, the definitions of the trigonometric functions and the standard identities for exponentials are sufficient to easily derive most trigonometric identities. It provides a powerful connection between analysis and trigonometry, and provides an interpretation of the sine and cosine functions as weighted sums of the exponential function: $$\begin{array}{cc}\cos x& =\mathrm{Re}\left(e^{ix}\right)=\frac{e^{ix}+e^{-ix}}{2},\\ \sin x& =\mathrm{Im}\left(e^{ix}\right)=\frac{e^{ix}-e^{-ix}}{2i}.\end{array}$$

The two equations above can be derived by adding or subtracting Euler's formulas: $$\begin{array}{cc}{e}^{ix}& =\cos x+i\sin x,\\ e^{-ix}& =\cos (-x)+i\sin (-x)=\cos x-i\sin x\end{array}$$ and solving for either cosine or sine.

These formulas can even serve as the definition of the trigonometric functions for complex arguments Template:Mvar. For example, letting Template:Math, we have: $$\begin{array}{cc}\cos iy& =\frac{e^{-y}+e^y}{2}=\cosh y,\\ \sin iy& =\frac{e^{-y}-e^y}{2i}=\frac{e^y-e^{-y}}{2}i=i\sinh y.\end{array}$$

In addition $$\begin{array}{cc}\cosh ix& =\frac{e^{ix}+e^{-ix}}{2}=\cos x,\\ \sinh ix& =\frac{e^{ix}-e^{-ix}}{2}=i\sin x.\end{array}$$

Complex exponentials can simplify trigonometry, because they are mathematically easier to manipulate than their sine and cosine components. One technique is simply to convert sines and cosines into equivalent expressions in terms of exponentials sometimes called complex sinusoids. After the manipulations, the simplified result is still real-valued. For example:

$$\begin{array}{cc}\cos x\cos y& =\frac{e^{ix}+e^{-ix}}{2}\cdot \frac{e^{iy}+e^{-iy}}{2}\\ & =\frac{1}{2}\cdot \frac{e^{i(x+y)}+e^{i(x-y)}+e^{i(-x+y)}+e^{i(-x-y)}}{2}\\ & =\frac{1}{2}(\frac{e^{i(x+y)}+e^{-i(x+y)}}{2}+\frac{e^{i(x-y)}+e^{-i(x-y)}}{2})\\ & =\frac{1}{2}(\cos (x+y)+\cos (x-y)).\end{array}$$

Another technique is to represent sines and cosines in terms of the real part of a complex expression and perform the manipulations on the complex expression. For example: $$\begin{array}{cc}\cos nx& =\mathrm{Re}\left(e^{inx}\right)\\ & =\mathrm{Re}(e^{i(n-1)x}\cdot e^{ix})\\ & =\mathrm{Re}(e^{i(n-1)x}\cdot (\underset{2\cos x}{\underbrace{e^{ix}+e^{-ix}}}-e^{-ix}\left)\right)\\ & =\mathrm{Re}(e^{i(n-1)x}\cdot 2\cos x-e^{i(n-2)x})\\ & =\cos [(n-1)x]\cdot [2\cos x]-\cos [(n-2)x].\end{array}$$

This formula is used for recursive generation of Template:Math for integer values of Template:Mvar and arbitrary Template:Mvar (in radians).

Considering Template:Math a parameter in equation above yields recursive formula for Chebyshev polynomials of the first kind.

Topological interpretation

Template:Unreferenced section In the language of topology, Euler's formula states that the imaginary exponential function $t\mapsto e^{it}$ is a (surjective) morphism of topological groups from the real line $ℝ$ to the unit circle ${𝕊}^1$. In fact, this exhibits $ℝ$ as a covering space of ${𝕊}^1$. Similarly, Euler's identity says that the kernel of this map is $\tau \mathbb{Z}$, where $\tau =2\pi$. These observations may be combined and summarized in the commutative diagram below:

Euler's formula and identity combined in diagrammatic form

Other applications

In differential equations, the function Template:Math is often used to simplify solutions, even if the final answer is a real function involving sine and cosine. The reason for this is that the exponential function is the eigenfunction of the operation of differentiation.

In electrical engineering, signal processing, and similar fields, signals that vary periodically over time are often described as a combination of sinusoidal functions (see Fourier analysis), and these are more conveniently expressed as the sum of exponential functions with imaginary exponents, using Euler's formula. Also, phasor analysis of circuits can include Euler's formula to represent the impedance of a capacitor or an inductor.

In the four-dimensional space of quaternions, there is a sphere of imaginary units. For any point Template:Mvar on this sphere, and Template:Mvar a real number, Euler's formula applies: $$\exp xr=\cos x+r\sin x,$$ and the element is called a versor in quaternions. The set of all versors forms a 3-sphere in the 4-space.

Other special cases

The special cases that evaluate to units illustrate rotation around the complex unit circle:

Template:Math	Template:Math
Template:Math	Template:Math
Template:Math	Template:Math
Template:Math	Template:Math
Template:Math	Template:Math

The special case at Template:Math (where Template:Math, one turn) yields Template:Math. This is also argued to link five fundamental constants with three basic arithmetic operations, but, unlike Euler's identity, without rearranging the addends from the general case: $$\begin{array}{cc}{e}^{i\tau }& =\cos \tau +i\sin \tau \\ & =1+0\end{array}$$ An interpretation of the simplified form Template:Math is that rotating by a full turn is an identity function.

See also

• Complex number
• Euler's identity
• Integration using Euler's formula
• History of Lorentz transformations
• List of topics named after Leonhard Euler

Further reading

• Template:Cite book
• Template:Cite book

External links

• Elements of Algebra