Euler's formula

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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 x,

$e^{ix} = \cos x + i\sin x \$

where e is the base of the natural logarithm, i is the imaginary unit, and cos and sin are the trigonometric functions cosine and sine respectively, with the argument x given in radians. This complex exponential function is sometimes denoted cis(x) ("cosine plus i sine"). The formula is still valid if x is a complex number, and so some authors refer to the more general complex version as Euler's formula.[1]

Euler's formula is ubiquitous in mathematics, physics, and engineering. The physicist Richard Feynman called the equation "our jewel" and "the most remarkable formula in mathematics."[2]

History

It was Johann Bernoulli who noted that[3]

$\frac{1}{1+x^2}=\frac{1}{2} \left(\frac{1}{1-ix}+\frac{1}{1+ix} \right) \ .$

And since

$\int \frac{dx}{1+ax}=\frac{1}{a}\ln(1+ax)+C \ ,$

the above equation tells us something about complex logarithms. 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 the complex logarithms can have infinitely many values.

Meanwhile, Roger Cotes, in 1714, discovered that

$\ln(\cos x + i\sin x)=ix \$

("ln" is the natural logarithm with base e).[4]

Cotes missed the fact that a complex logarithm can have infinitely many values, differing by multiples of 2π, due to the periodicity of the trigonometric functions.

Around 1740 Euler turned his attention to the exponential function instead of logarithms, and obtained the formula used today that is named after him. It was published in 1748, obtained by comparing the series expansions of the exponential and trigonometric expressions.[4]

None of these mathematicians saw the geometrical interpretation of the formula; the view of complex numbers as points in the complex plane was described some 50 years later by Caspar Wessel.

Applications in complex number theory

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

The original proof is based on the Taylor series expansions of the exponential function ez (where z is a complex number) and of sin x and cos x for real numbers x (see below). In fact, the same proof shows that Euler's formula is even valid for all complex numbers x.

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 $z = x + iy$, and its complex conjugate, $\bar{z} = x - iy$, can be written as

\begin{align} z & = x + iy & = |z| (\cos \phi + i\sin \phi ) & = r e^{i \phi} \\ \bar{z} & = x - iy & = |z| (\cos \phi - i\sin \phi ) & = r e^{-i \phi} \end{align}

where

$x = \mathrm{Re}\{z\} \,$ the real part
$y = \mathrm{Im}\{z\} \,$ the imaginary part
$r = |z| = \sqrt{x^2+y^2}$ the magnitude of z
$\phi = \arg z = \,$ atan2(y, x) .

$\phi \,$ is the argument of z—i.e., the angle between the x axis and the vector z measured counterclockwise and in radians—which is defined up to addition of 2π. Many texts write θ = tan−1(y/x) instead of θ = atan2(y,x), but the first equation needs adjustment when x ≤ 0. This is because, for any real x, y not both zero, the angles of the vectors (x,y) and (-x,-y) differ by π radians, but have the identical value of tan(θ) = y/x.

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) that

$a = e^{\ln (a)} \$

and that

$e^a e^b = e^{a + b} \$

both valid for any complex numbers a and b.

Therefore, one can write:

$z = |z| e^{i \phi} = e^{\ln |z|} e^{i \phi} = e^{\ln |z| + i \phi} \$

for any z ≠ 0. Taking the logarithm of both sides shows that:

$\ln z= \ln |z| + i \phi \ .$

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 $\phi$ is multi-valued.

Finally, the other exponential law

$(e^a)^k = e^{a k} \ ,$

which can be seen to hold for all integers k, 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 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{align} \cos x & = \mathrm{Re}\{e^{ix}\} ={e^{ix} + e^{-ix} \over 2} \\ \sin x & = \mathrm{Im}\{e^{ix}\} ={e^{ix} - e^{-ix} \over 2i} \end{align}

The two equations above can be derived by adding or subtracting Euler's formulas:

\begin{align} e^{ix} & = \cos x + i \sin x \; \\ e^{-ix} & = \cos(- x) + i \sin(- x) = \cos x - i \sin x \; \end{align}

and solving for either cosine or sine.

These formulas can even serve as the definition of the trigonometric functions for complex arguments x. For example, letting x = iy, we have:

\begin{align} \cos(iy) & = {e^{-y} + e^{y} \over 2} = \cosh(y) \\ \sin(iy) & = {e^{-y} - e^{y} \over 2i} = - {e^{y} - e^{-y} \over 2i} = i\sinh(y) \ . \end{align}

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

\begin{align} \cos x\cdot \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} \bigg[ \underbrace{ \frac{e^{i(x+y)} + e^{-i(x+y)}}{2} }_{\cos(x+y)} + \underbrace{ \frac{e^{i(x-y)} + e^{-i(x-y)}}{2} }_{\cos(x-y)} \bigg] \ \end{align}

Another technique is to represent the sinusoids in terms of the real part of a more complex expression, and perform the manipulations on the complex expression. For example:

\begin{align} \cos(nx) & = \mathrm{Re} \{\ e^{inx}\ \} = \mathrm{Re} \{\ e^{i(n-1)x}\cdot e^{ix}\ \} \\ & = \mathrm{Re} \{\ e^{i(n-1)x}\cdot (\underbrace{e^{ix} + e^{-ix}}_{2\cos(x)} - e^{-ix})\ \} \\ & = \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{align}

This formula is used for recursive generation of cos(nx) for integer values of n and arbitrary x (in radians).

Topological interpretation

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 $\mathbb R$ to the unit circle $\mathbb S^1$. In fact, this exhibits $\mathbb R$ as a covering space of $\mathbb S^1$. 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:

Other applications

In differential equations, the function eix is often used to simplify derivations, even if the final answer is a real function involving sine and cosine. The reason for this is that the complex exponential is the eigenfunction of differentiation. Euler's identity is an easy consequence of Euler's formula.

In electronic engineering and other fields, signals that vary periodically over time are often described as a combination of sine and cosine functions (see Fourier analysis), and these are more conveniently expressed as the real part 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.

Definitions of complex exponentiation

The exponential function ex for real values of x 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 ez for complex values of z simply by substituting z in place of x and using the complex algebraic operations. In particular we may use either of the two following definitions which are equivalent. From a more advanced perspective, each of these definitions may be interpreted as giving the unique analytic continuation of ex to the complex plane.

Power series definition

For complex z

$e^z = 1 + \frac{z}{1!} + \frac{z^2}{2!} + \frac{z^3}{3!} + \dots = \sum_{n=0}^{\infty} \frac{z^n}{n!}.$

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

Limit definition

For complex z

$e^z = \lim_{n \rightarrow \infty} \left(1+\frac{z}{n}\right)^n ~.$

Proofs

Various proofs of the formula are possible.

Using power series

Here is a proof of Euler's formula using power series expansions as well as basic facts about the powers of i:[5]

\begin{align} i^0 &{}= 1, \quad & i^1 &{}= i, \quad & i^2 &{}= -1, \quad & i^3 &{}= -i, \\ i^4 &={} 1, \quad & i^5 &={} i, \quad & i^6 &{}= -1, \quad & i^7 &{}= -i, \end{align}

and so on. Using now the power series definition from above we see that for real values of x

\begin{align} 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 \\[8pt] &{}= 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!} + \cdots \\[8pt] &{}= \left( 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \frac{x^8}{8!} - \cdots \right) + i\left( x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \right) \\[8pt] &{}= \cos x + i\sin x \ . \end{align}

In the last step we have simply recognized the Maclaurin series for cos(x) and sin(x). The rearrangement of terms is justified because each series is absolutely convergent.

Using the limit definition

The exponential function ez can be defined as the limit of (1 + z/n)n, as n approaches infinity. In this animation, z=/3, and n takes various increasing values from 1 to 100. The computation of (1 + z/n)n is displayed as the combined effect of n repeated multiplications in the complex plane. As n gets larger, the points approach the complex unit circle (dashed line), covering an angle of π/3 radians.

An alternative proof[6] is based on the limit definition of $e^z$:

$e^z = \lim_{n\rightarrow\infty} \left(1+\frac{z}{n}\right)^n$.

Substitute $z=ix$, and let n be a very large integer, say 1000. Then, based on the limit definition, the complex number (1+ix/1000)1000 is supposed to be a good approximation to eix. So, what is the value of (1+ix/1000)1000?

Consider the sequence of 1000 complex numbers:

$1, \, \left(1+\frac{ix}{1000}\right), \, \left(1+\frac{ix}{1000}\right)^2, \ldots, \, \left(1+\frac{ix}{1000}\right)^{1000}$

(We started with 1, and successively multiplied it by (1+ix/1000), 1000 times.) If the points of this sequence are plotted in the complex plane (see animation at right), they approximately trace out the unit circle, with each point being x/1000 radians counterclockwise of the previous point. (The proof of this is based on the rules of trigonometry and complex-number algebra.[6]) Therefore, the last point in the sequence, (1 + ix/1000)1000, is approximately the point on the unit circle of the complex plane located x radians counterclockwise from +1, that is the point cos x + i sin x. If we replaced the number 1000 by larger and larger numbers, all of the approximations in this paragraph become more and more accurate. Therefore, eix = cos x + i sin x.

Using calculus

Another proof[7] is based on the fact that all complex numbers can be expressed in polar coordinates. Therefore for some $r$ and $\theta$ depending on $x$,

$e^{ix} = r (\cos(\theta) + i \sin(\theta))\,.$

Now from any of the definitions of the exponential function it can be shown that the derivative of $e ^{ix}$ is $i e ^{ix}$. 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 $r (\cos(\theta) + i \sin(\theta))$ for $e^{ix}$ and equating real and imaginary parts in this formula gives $\textstyle \frac{dr}{dx} = 0$ and $\textstyle \frac{d\theta}{dx} = 1$. Together with the initial values $r(0) = 1$ and $\theta(0) = 0$ which come from $e^{i0} = 1$ this gives $r=1$ and $\theta=x$. This proves the formula $e^{ix} = 1(\cos(x)+i \sin(x))$.