Euler's formula

This article is about Euler's formula in complex analysis. For other uses, see List of things named after Leonhard Euler § Formulae.

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, one has

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

where e is the base of the natural logarithm, i is the imaginary unit, and <span class="texhtml " {{#if:cos and <span class="texhtml " {{#if:sin are the trigonometric functions cosine and sine respectively. This complex exponential function is sometimes denoted <span class="texhtml " {{#if:cis x ("cosine plus i sine"). The formula is still valid if x 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 <span class="texhtml " {{#if:x = π, Euler's formula may be rewritten as <span class="texhtml " {{#if:e^iπ + 1 = 0 or <span class="texhtml " {{#if:e^iπ = -1, 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 <span class="texhtml " {{#if:2πi.

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

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

And since

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

the above equation tells us something about complex logarithms 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

Further information: Exponentiation § Complex exponents with a positive real base, and Exponential function § On the complex plane

The exponential function <span class="texhtml " {{#if:e^x 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 <span class="texhtml " {{#if:e^z for complex values of z simply by substituting z in place of x 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 <span class="texhtml " {{#if:e^x 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 z

$$e^{z}=1+{\frac {z}{1!}}+{\frac {z^{2}}{2!}}+{\frac {z^{3}}{3!}}+\cdots =\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}.$$

Using the ratio test, it is possible to show that this power series has an infinite radius of convergence and so defines <span class="texhtml " {{#if:e^z for all complex z.

Limit definition

For complex z

$$e^{z}=\lim _{n\to \infty }\left(1+{\frac {z}{n}}\right)^{n}.$$

Here, n is restricted to positive integers, so there is no question about what the power with exponent n 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 <span class="texhtml " {{#if:f(θ)

$$f(\theta )={\frac {\cos \theta +i\sin \theta }{e^{i\theta }}}=e^{-i\theta }\left(\cos \theta +i\sin \theta \right)$$

for real θ. Differentiating gives by the product rule

$$f'(\theta )=e^{-i\theta }\left(i\cos \theta -\sin \theta \right)-ie^{-i\theta }\left(\cos \theta +i\sin \theta \right)=0$$

Thus, <span class="texhtml " {{#if:f(θ) is a constant. Since <span class="texhtml " {{#if:f(0) = 1, then <span class="texhtml " {{#if:f(θ) = 1 for all real θ, and thus

$$e^{i\theta }=\cos \theta +i\sin \theta .$$

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:

$${\begin{aligned}i^{0}&=1,&i^{1}&=i,&i^{2}&=-1,&i^{3}&=-i,\\i^{4}&=1,&i^{5}&=i,&i^{6}&=-1,&i^{7}&=-i\\&\vdots &&\vdots &&\vdots &&\vdots \end{aligned}}$$

Using now the power-series definition from above, we see that for real values of x

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

where in the last step we recognize the two terms are the Maclaurin series for <span class="texhtml " {{#if:cos x and <span class="texhtml " {{#if:sin x. 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 r and θ depending on x,

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

No assumptions are being made about r and θ; 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 <span class="texhtml " {{#if:e^ix is <span class="texhtml " {{#if:ie^ix. Therefore, differentiating both sides gives

$$ie^{ix}=\left(\cos \theta +i\sin \theta \right){\frac {dr}{dx}}+r\left(-\sin \theta +i\cos \theta \right){\frac {d\theta }{dx}}.$$

Substituting <span class="texhtml " {{#if:r(cos θ + i sin θ) for <span class="texhtml " {{#if:e^ix and equating real and imaginary parts in this formula gives <span class="texhtml " {{#if:dr/dx = 0 and <span class="texhtml " {{#if:dθ/dx = 1. Thus, r is a constant, and θ is <span class="texhtml " {{#if:x + C for some constant C. The initial values <span class="texhtml " {{#if:r(0) = 1 and <span class="texhtml " {{#if:θ(0) = 0 come from <span class="texhtml " {{#if:e⁰ⁱ = 1, giving <span class="texhtml " {{#if:r = 1 and <span class="texhtml " {{#if:θ = x. 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 <span class="texhtml " {{#if:e^iφ = cos φ + i sin φ illustrated in the complex plane.

Interpretation of the formula

This formula can be interpreted as saying that the function <span class="texhtml " {{#if:e^iφ is a unit complex number, i.e., it traces out the unit circle in the complex plane as φ ranges through the real numbers. Here φ 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 <span class="texhtml " {{#if:e^z (where z is a complex number) and of <span class="texhtml " {{#if:sin x and <span class="texhtml " {{#if:cos x for real numbers x (see above). 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 <span class="texhtml " {{#if:z = x + iy, and its complex conjugate, <span class="texhtml " {{#if:z = x − iy, can be written as

$${\begin{aligned}z&=x+iy=|z|(\cos \varphi +i\sin \varphi )=re^{i\varphi },\\{\bar {z}}&=x-iy=|z|(\cos \varphi -i\sin \varphi )=re^{-i\varphi },\end{aligned}}$$

where

• <span class="texhtml " {{#if:x = Re z is the real part,
• <span class="texhtml " {{#if:y = Im z is the imaginary part,
• <span class="texhtml " {{#if:r = |z| = √x² + y² is the magnitude of z and
• <span class="texhtml " {{#if:φ = arg z = atan2(y, x).

φ is the argument of z, i.e., the angle between the x axis and the vector z measured counterclockwise in radians, which is defined up to addition of <span class="texhtml " {{#if:2π. Many texts write <span class="texhtml " {{#if:φ = tan⁻¹ y/x instead of <span class="texhtml " {{#if:φ = atan2(y, x), but the first equation needs adjustment when <span class="texhtml " {{#if:x ≤ 0. This is because for any real x and y, not both zero, the angles of the vectors <span class="texhtml " {{#if:(x, y) and <span class="texhtml " {{#if:(−x, −y) differ by π radians, but have the identical value of <span class="texhtml " {{#if:tan φ = y/x.

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 a and b. Therefore, one can write:

$$z=\left|z\right|e^{i\varphi }=e^{\ln \left|z\right|}e^{i\varphi }=e^{\ln \left|z\right|+i\varphi }$$

for any <span class="texhtml " {{#if:z ≠ 0. Taking the logarithm of both sides shows that

$$\ln z=\ln \left|z\right|+i\varphi ,$$

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 φ is multi-valued.

Finally, the other exponential law

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

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, 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:

$${\begin{aligned}\cos x&=\operatorname {Re} \left(e^{ix}\right)={\frac {e^{ix}+e^{-ix}}{2}},\\\sin x&=\operatorname {Im} \left(e^{ix}\right)={\frac {e^{ix}-e^{-ix}}{2i}}.\end{aligned}}$$

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

$${\begin{aligned}e^{ix}&=\cos x+i\sin x,\\e^{-ix}&=\cos(-x)+i\sin(-x)=\cos x-i\sin x\end{aligned}}$$

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 <span class="texhtml " {{#if:x = iy, we have:

$${\begin{aligned}\cos iy&={\frac {e^{-y}+e^{y}}{2}}=\cosh y,\\\sin iy&={\frac {e^{-y}-e^{y}}{2i}}={\frac {e^{y}-e^{-y}}{2}}i=i\sinh y.\end{aligned}}$$

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:

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

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

$${\begin{aligned}\cos nx&=\operatorname {Re} \left(e^{inx}\right)\\&=\operatorname {Re} \left(e^{i(n-1)x}\cdot e^{ix}\right)\\&=\operatorname {Re} {\Big (}e^{i(n-1)x}\cdot {\big (}\underbrace {e^{ix}+e^{-ix}} _{2\cos x}-e^{-ix}{\big )}{\Big )}\\&=\operatorname {Re} \left(e^{i(n-1)x}\cdot 2\cos x-e^{i(n-2)x}\right)\\&=\cos[(n-1)x]\cdot [2\cos x]-\cos[(n-2)x].\end{aligned}}$$

This formula is used for recursive generation of <span class="texhtml " {{#if:cos nx for integer values of n and arbitrary x (in radians).

See also: Phasor § Arithmetic

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

See also: Complex number § Applications

In differential equations, the function <span class="texhtml " {{#if:e^ix 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 r on this sphere, and x 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.

See also

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

Further reading

• Nahin, Paul J. (2006). Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills. Princeton University Press. ISBN 978-0-691-11822-2.
• Wilson, Robin (2018). Euler's Pioneering Equation: The Most Beautiful Theorem in Mathematics. Oxford: Oxford University Press. ISBN 978-0-19-879492-9. MR 3791469.

External links

• Elements of Algebra