- This article is about Euler's formula in complex analysis. For Euler's formula in graph theory and polyhedral combinatorics see Euler characteristic. See also topics named after Euler.
Euler's formula states that, for any real number x,
- Failed to parse (Missing texvc executable; please see math/README to configure.): 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 trigonometric functions (here it is assumed that, when calculating the sine and cosine, x is measured in radians rather than in degrees). 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" and "the most remarkable formula in mathematics".[2]
History
Euler's formula was proven for the first time by Roger Cotes in 1714 in the form
- Failed to parse (Missing texvc executable; please see math/README to configure.): \ln(\cos(x) + i\sin(x))=ix \
(where "ln" means natural logarithm, i.e. log with base e).[3]
It was Euler who published the equation in its current form in 1748, basing his proof 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). Euler considered it natural to introduce students to complex numbers much earlier than we do today. In his elementary algebra text book, Elements of Algebra, he introduces these numbers almost at once and then uses them in a natural way throughout.
Applications in complex number theory
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function. (Euler's identity is a special case of the Euler formula.)
This formula can be interpreted as saying that the function eix 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 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 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
- Failed to parse (Missing texvc executable; please see math/README to configure.): z = x + iy = |z| (\cos \phi + i\sin \phi ) = |z| e^{i \phi} \,
- Failed to parse (Missing texvc executable; please see math/README to configure.): \bar{z} = x - iy = |z| (\cos \phi - i\sin \phi ) = |z| e^{-i \phi} \,
where
- Failed to parse (Missing texvc executable; please see math/README to configure.): x = \mathrm{Re}\{z\} \,
the real part
- Failed to parse (Missing texvc executable; please see math/README to configure.): y = \mathrm{Im}\{z\} \,
the imaginary part
- Failed to parse (Missing texvc executable; please see math/README to configure.): |z| = \sqrt{x^2+y^2}
the magnitude of z
and Failed to parse (Missing texvc executable; please see math/README to configure.): \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.
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
- Failed to parse (Missing texvc executable; please see math/README to configure.): a = e^{\ln (a)}\,
and that
- Failed to parse (Missing texvc executable; please see math/README to configure.): e^a e^b = e^{a + b}\,
both valid for any complex numbers a and b.
Therefore, one can write:
- Failed to parse (Missing texvc executable; please see math/README to configure.): z = |z| e^{i \phi} = e^{\ln |z|} e^{i \phi} = e^{\ln |z| + i \phi}\,
for any Failed to parse (Missing texvc executable; please see math/README to configure.): z\ne 0 . Taking the logarithm of both sides shows that:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \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, due to the fact that Failed to parse (Missing texvc executable; please see math/README to configure.): \phi \,
is multi-valued.
Finally, the other exponential law
- Failed to parse (Missing texvc executable; please see math/README to configure.): (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:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \cos x = \mathrm{Re}\{e^{ix}\} ={e^{ix} + e^{-ix} \over 2}
- Failed to parse (Missing texvc executable; please see math/README to configure.): \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:
- Failed to parse (Missing texvc executable; please see math/README to configure.): e^{ix} = \cos x + i \sin x \;
- Failed to parse (Missing texvc executable; please see math/README to configure.): 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:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \cos(iy) = {e^{-y} + e^{y} \over 2} = \cosh(y)
- Failed to parse (Missing texvc executable; please see math/README to configure.): \sin(iy) = {e^{-y} - e^{y} \over 2i} = i\cdot \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:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} \cos(x)\cdot \cos(y) & = \frac{(e^{ix}+e^{-ix})}{2} \cdot \frac{(e^{iy}+e^{-iy})}{2} \\ & = \frac{e^{i(x+y)}+e^{i(x-y)}+e^{i(-x+y)}+e^{i(-x-y)}}{4} \\ & = \frac{e^{i(x+y)}+e^{i(-x-y)}}{4}+\frac{e^{i(x-y)}+e^{i(-x+y)}}{4} \\ & = \frac{\cos(x+y)}{2} + \frac{\cos(x-y)}{2}. \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:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} \cos(x\cdot n)+\cos(x\cdot(n-2)) & = \mathrm{Re} \{\quad e^{ix n}+e^{ix(n-2)}\quad \} \\ & = \mathrm{Re} \{\quad e^{ix(n-1)}\cdot (e^{ix}+e^{-ix})\quad \} \\ & = \mathrm{Re} \{\quad e^{ix(n-1)}\cdot 2\cos(x)\quad \} \\ & = \cos(x\cdot(n-1))\cdot 2\cos(x). \end{align}
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. 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.
Proofs
Using Taylor series
Here is a proof of Euler's formula using Taylor series expansions as well as basic facts about the powers of i:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \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. The functions ex, cos(x) and sin(x) (assuming x is real) can be expressed using their Taylor expansions around zero:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} e^x &{}= 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots \\ \cos x &{}= 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots \\ \sin x &{}= x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \end{align}
For complex z we define each of these functions by the above series, replacing x with z. This is possible because the radius of convergence of each series is infinite. We then find that
- Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} e^{iz} &{}= 1 + iz + \frac{(iz)^2}{2!} + \frac{(iz)^3}{3!} + \frac{(iz)^4}{4!} + \frac{(iz)^5}{5!} + \frac{(iz)^6}{6!} + \frac{(iz)^7}{7!} + \frac{(iz)^8}{8!} + \cdots \\ &{}= 1 + iz - \frac{z^2}{2!} - \frac{iz^3}{3!} + \frac{z^4}{4!} + \frac{iz^5}{5!} - \frac{z^6}{6!} - \frac{iz^7}{7!} + \frac{z^8}{8!} + \cdots \\ &{}= \left( 1 - \frac{z^2}{2!} + \frac{z^4}{4!} - \frac{z^6}{6!} + \frac{z^8}{8!} - \cdots \right) + i\left( z - \frac{z^3}{3!} + \frac{z^5}{5!} - \frac{z^7}{7!} + \cdots \right) \\ &{}= \cos (z) + i\sin (z) \end{align}
The rearrangement of terms is justified because each series is absolutely convergent. Taking z = x to be a real number gives the original identity as Euler discovered it.
Using calculus
Define the (possibly complex) function f(x), of real variable x, as
- Failed to parse (Missing texvc executable; please see math/README to configure.): f(x) = \frac{\cos x + i\sin x}{e^{ix}}. \
Division by zero is precluded since the equation
- Failed to parse (Missing texvc executable; please see math/README to configure.): e^{ix} \cdot e^{-ix} = e^{ix \, + \, (-ix)} = e^0 = 1 \
implies that Failed to parse (Missing texvc executable; please see math/README to configure.): e^{ix} \
is never zero.
The derivative of f(x), according to the quotient rule, is:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} \frac{d}{dx}f(x) &{}= \frac{e^{ix} \cdot \frac{d}{dx}(\cos x+i\sin x) - (\cos x+i\sin x) \cdot \frac{d}{dx}(e^{ix})}{(e^{ix})^2} \\ &{}= \frac{e^{ix} \cdot (-\sin x + i\cos x) - (\cos x+i\sin x) \cdot (i e^{ix})}{(e^{ix})^2} \\ &{}= \frac{-\sin x \cdot e^{ix} - i^2 \sin x \cdot e^{ix}}{(e^{ix})^2} \quad \quad \quad (i^2=-1) \\ &{}= \frac{-\sin x \cdot e^{ix} + \sin x \cdot e^{ix}}{(e^{ix})^2} \\ &{}= 0. \end{align}
Therefore, f(x) must be a constant function in x. Because f(0) is known, the constant that f(x) equals for all real x is also known. Thus,
- Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\cos x + i \sin x}{e^{ix}} = f(x) = f(0) = \frac{\cos 0 + i \sin 0}{e^0} = 1 .
Rearranging, it follows that
- Failed to parse (Missing texvc executable; please see math/README to configure.): e^{ix} = \cos x + i \sin x \ .
Q.E.D.
Using ordinary differential equations
Define the function g(x) by
- Failed to parse (Missing texvc executable; please see math/README to configure.): g(x) \ \stackrel{\mathrm{def}}{=}\ e^{ix} .\
Considering that i is constant, the first and second derivatives of g(x) are
- Failed to parse (Missing texvc executable; please see math/README to configure.): g'(x) = i e^{ix} \
- Failed to parse (Missing texvc executable; please see math/README to configure.): g''(x) = i^2 e^{ix} = -e^{ix} \
because i 2 = −1 by definition. From this the following 2nd-order linear ordinary differential equation is constructed:
- Failed to parse (Missing texvc executable; please see math/README to configure.): g''(x) = -g(x) \
or
- Failed to parse (Missing texvc executable; please see math/README to configure.): g''(x) + g(x) = 0. \
Being a 2nd-order differential equation, there are two linearly independent solutions that satisfy it:
- Failed to parse (Missing texvc executable; please see math/README to configure.): g_1(x) = \cos(x) \
- Failed to parse (Missing texvc executable; please see math/README to configure.): g_2(x) = \sin(x). \
Both cos(x) and sin(x) are real functions in which the 2nd derivative is identical to the negative of that function. Any linear combination of solutions to a homogeneous differential equation is also a solution. Then, in general, the solution to the differential equation is
-
| Failed to parse (Missing texvc executable; please see math/README to configure.): g(x)\, |
Failed to parse (Missing texvc executable; please see math/README to configure.): = A g_1(x) + B g_2(x) \ |
|
Failed to parse (Missing texvc executable; please see math/README to configure.): = A \cos(x) + B \sin(x) \ |
for any constants A and B. But not all values of these two constants satisfy the known initial conditions for g(x):
- Failed to parse (Missing texvc executable; please see math/README to configure.): g(0) = e^{i0} = 1 \
- Failed to parse (Missing texvc executable; please see math/README to configure.): g'(0) = i e^{i0} = i \
.
However these same initial conditions (applied to the general solution) are
- Failed to parse (Missing texvc executable; please see math/README to configure.): g(0) = A \cos(0) + B \sin(0) = A \
- Failed to parse (Missing texvc executable; please see math/README to configure.): g'(0) = -A \sin(0) + B \cos(0) = B \
resulting in
- Failed to parse (Missing texvc executable; please see math/README to configure.): g(0) = A = 1 \
- Failed to parse (Missing texvc executable; please see math/README to configure.): g'(0) = B = i \
and, finally,
- Failed to parse (Missing texvc executable; please see math/README to configure.): g(x) \ \stackrel{\mathrm{def}}{=}\ e^{ix} = \cos(x) + i \sin(x). \
Q.E.D.
See also
References
- ^ Moskowitz, Martin A. (2002). A Course in Complex Analysis in One Variable. World Scientific Publishing Co., p. 7. ISBN 981-02-4780-X.
- ^ Feynman, Richard P. (1977). The Feynman Lectures on Physics, vol. I. Addison-Wesley, p. 22-10. ISBN 0-201-02010-6.
- ^ John Stillwell (2002). Mathematics and Its History. Springer.
External links
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