Euler's formula

This article is about Euler's formula in complex analysis. For Euler's formula in algebraic topology and polyhedral combinatorics see Euler characteristic. See also topics named after Euler.

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The mathematical constant e

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People John Napier · Leonhard Euler

Schanuel's conjecture

Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that demonstrates the deep 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 called cis(x). 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]

Richard Feynman called Euler's formula "our jewel"^[2] and "one of the most remarkable, almost astounding, formulas in all of mathematics."^[3]

1 History
2 Applications in complex number theory
3 Relationship to trigonometry
4 Other applications
5 Definitions of complex exponentiation
- 5.1 Power series definition
- 5.2 Limit definition
6 Proofs
7 See also
8 References
9 External links

History

It was Bernoulli [1702] who noted that

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

And since

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

the above equation tells us something about complex logarithms. Bernoulli, however, did not evaluate the integral. His correspondence with Euler (who also knew the above equation) shows that he didn't fully understand logarithms. Euler also suggested that the complex logarithms can have infinitely many values.

Meanwhile, Roger Cotes, in 1714, discovered

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

(where "ln" means natural logarithm, i.e. log with base e).^[4] We now know that the above equation is only true modulo integer multiples of $2\pi i$ , but Cotes missed the fact that a complex logarithm can have infinitely many values which owes to the periodicity of the trigonometric functions.

It was Euler (presumably around 1740) who turned his attention to the exponential function instead of logarithms, and obtained the correct formula now coined after his name. It was published in 1748, and his proof was based on the infinite series of both sides being equal. Neither of these men saw the geometrical interpretation of the formula: the view of complex numbers as points in the complex plane arose only some 50 years later (see Caspar Wessel).

Applications in complex number theory

Three-dimensional vizualization of Euler's formula

This formula can be interpreted as saying that the function e^ix traces out the unit circle in the complex number 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 e^z (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 z.

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 reduces the number of terms from two to one, which simplifies the mathematics when used in multiplication or powers of complex numbers. Any complex number z = x + iy can be written as

$z = x + iy = |z| (\cos \phi + i\sin \phi ) = r e^{i \phi} \,$

$\bar{z} = x - iy = |z| (\cos \phi - i\sin \phi ) = r e^{-i \phi} \,$

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 tan^-1(y/x) instead of atan2(y,x) but this needs adjustment when x ≤ 0.

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

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:

$\cos x = \mathrm{Re}\{e^{ix}\} ={e^{ix} + e^{-ix} \over 2}$

$\sin x = \mathrm{Im}\{e^{ix}\} ={e^{ix} - e^{-ix} \over 2i}.$

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

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

$e^{-ix} = \cos(- x) + i \sin(- x) = \cos x - i \sin x \;$

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:

$\cos(iy) = {e^{-y} + e^{y} \over 2} = \cosh(y)$

$\sin(iy) = {e^{-y} - e^{y} \over 2i} = -{1 \over i} {e^{y} - e^{-y} \over 2} = i\sinh(y).$

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:

$\begin{align} \cos x\cdot \cos y & = \frac{(e^{ix}+e^{-ix})}{2} \cdot \frac{(e^{iy}+e^{-iy})}{2} \\ & = \frac{1}{2}\cdot \frac{e^{i(x+y)}+e^{i(x-y)}+e^{i(-x+y)}+e^{i(-x-y)}}{2} \\ & = \frac{1}{2} \left[ \underbrace{ \frac{e^{i(x+y)} + e^{-i(x+y)}}{2} }_{\cos(x+y)} + \underbrace{ \frac{e^{i(x-y)} + e^{-i(x-y)}}{2} }_{\cos(x-y)} \right]. \end{align}$

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:

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

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

Other applications

In differential equations, the function e^ix 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 electrical 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

Main articles: Exponentiation and Exponential function

The exponential function 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 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 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 e^x to the complex plane.

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 e^z 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:

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

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

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

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

Using calculus

Several other proofs are based on the following identity obtained by differentiating the power series definition of e^ix. Indeed, since this series converges absolutely for all complex numbers we can differentiate it term by term to obtain

$\begin{align} \frac{\mathrm{d}}{\mathrm{d}x} e^{ix} & = \frac{\mathrm{d}}{\mathrm{d}x} \sum_{n=0}^{\infty} \frac{(ix)^n}{n!} \\ & = i\sum_{n=1}^\infty i^{n-1} \frac{x^{n-1}}{(n-1)!} \\ & = i e^{ix}. \end{align}$

Now we define the function

$\begin{align} f(x) = (\cos x - i \sin x) \cdot e^{ix}. \end{align}$

The derivative of ƒ(x) according to the product rule (note that the product rule can be proved to hold for complex valued functions of a real variable using precisely the same proof as in the real case) is:

$\begin{align} \frac{d}{dx}f(x) &{}= (\cos x - i\sin x)\cdot\frac{d}{dx}e^{ix} + \frac{d}{dx}(\cos x - i\sin x)\cdot e^{ix} \\ &{}= (\cos x - i\sin x)(i e^{ix}) + (-\sin x - i\cos x)\cdot e^{ix} \\ &{}= (i\cos x + \sin x - \sin x - i\cos x)\cdot e^{ix} \\ &{}= 0. \end{align}$

Therefore, ƒ(x) must be a constant function in x. Because ƒ(0)=1 in fact ƒ(x) = 1 for all x , and so multiplying by cos x + i sin x, we get

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

Using differential equations

Here is another proof that follows from the differential identity above. Define a new function ƒ(x) of the real variable x as

$\begin{align} f(x) = \cos x + i \sin x. \end{align}$

Then we may check that

$\begin{align} \frac{\mathrm{d}f}{\mathrm{d}x}(x) & = -\sin x + i \cos x \\ & = i f(x). \end{align}$

Thus ƒ(x) and e^ix satisfy the same system of ordinary differential equations (here the complex values are considered as points in the plane ℝ²). If we also note that both functions are equal to 1 at x = 0, then by the uniqueness of solutions to ordinary differential equations (see Picard–Lindelöf theorem and note the comments concerning global uniqueness in the proof section there) they must be equal everywhere.

References

↑ Moskowitz, Martin A. (2002). A Course in Complex Analysis in One Variable. World Scientific Publishing Co.. pp. 7. ISBN 981-02-4780-X.
↑ Feynman, Richard P. (1977). The Feynman Lectures on Physics, vol. I. Addison-Wesley. pp. 22–10. ISBN 0-201-02010-6.
↑ Feynman, Richard P. (1977). The Feynman Lectures on Physics, vol. I. Addison-Wesley. pp. 22–1. ISBN 0-201-02010-6.
↑ John Stillwell (2002). Mathematics and Its History. Springer.