Euler's formula

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 \[e^{i x} = \cos x + i \sin x,\] 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. 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 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 x = π, Euler's formula may be rewritten as e = −1 or e + 1 = 0, which is known as Euler's identity.

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 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.

The exponential function e 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 e 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 e to the complex plane.

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 \[f(0) = 1.\]

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 e for all complex z.

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.

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

Consider the function 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, f(θ) is a constant. Since the exponential function is 1 for θ = 0, by definition, and the complex trig function also evaluates to 1 there, f(0) = 1/1 = 1, then f(θ) = 1 for all real θ, and thus \[e^{i\theta} = \cos\theta + i\sin\theta.\]

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 cos x and sin x. The splitting into two series is justified since we know the series for sine and cosine converge.

Another proof is based on the fact that all complex numbers can be expressed in polar coordinates and on the assumption that \(e^{ix}\) can be likewise represented; this will be the case if we find a solution. Therefore, for some r and θ depending on x, \[e^{i x} = r \left(\cos \theta + i \sin \theta\right).\] 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 e is ie. Therefore, differentiating both sides gives \[i e ^{ix} = \left(\cos \theta + i \sin \theta\right) \frac{dr}{dx} + r \left(-\sin \theta + i \cos \theta\right) \frac{d\theta}{dx}.\] Substituting r(cos θ + i sin θ) for e and equating real and imaginary parts in this formula gives ⁠dr/dx⁠ = 0 and ⁠dθ/dx⁠ = 1. Thus, r is a constant, and θ is x + C for some constant C. We now have \[e^{i\theta}=r e^{iC}(\cos \theta +i \sin \theta);\] knowing that e = 1, for θ = 0, this becomes \[1=r e^{iC}(\cos(0)+i\sin(0))=r e^{iC}(1+i0)=r e^{iC},\] giving us the constant \(r e^{iC}=1\) and proving the formula \[e^{i \theta} = \cos \theta + i \sin \theta.\]

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:

In differential equations, the function e 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.