Euler's formula
Euler's formula is the identity
[ e^{ix}=\cos x+i\sin x, ]
where (e) is the base of the natural logarithm, (i) is the imaginary unit, and (x) is a real number measured in radians. The identity connects the complex exponential function with the trigonometric functions and provides an analytic description of rotation in the complex plane. Its extension
[ e^{iz}=\cos z+i\sin z ]
holds for every complex number (z), when the exponential, sine, and cosine functions are defined by their corresponding power series.
The formula takes its name from Leonhard Euler, who presented it in substantially modern form in his 1748 work Introductio in analysin infinitorum. Earlier relations established by Roger Cotes and Abraham_de Moivre contained closely related logarithmic and trigonometric structures. Euler's notation unified these results within a general theory of complex exponentiation.
Mathematical formulation
For a real variable (x), the exponential function has the series expansion
[ e^{ix}
\sum_{n=0}^{\infty}\frac{(ix)^n}{n!}. ]
The powers of (i) repeat with period four:
[ i^{4k}=1,\qquad i^{4k+1}=i,\qquad i^{4k+2}=-1,\qquad i^{4k+3}=-i. ]
Separating the even-indexed and odd-indexed terms therefore gives
[ e^{ix}
\left( 1-\frac{x^2}{2!}+\frac{x^4}{4!}-\cdots \right) +i\left( x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots \right). ]
The two parenthesized series are the Maclaurin series for cosine and sine, respectively. Consequently,
[ e^{ix}=\cos x+i\sin x. ]
Because all three power series converge for every complex argument, the same calculation establishes the identity throughout the complex plane. This series-based derivation does not depend on a prior geometric definition of trigonometric functions and accordingly belongs to the analytic theory of entire functions.
An equivalent characterization follows from differential equations. The function (f(x)=e^{ix}) satisfies
[ f'(x)=if(x),\qquad f(0)=1. ]
The function (g(x)=\cos x+i\sin x) satisfies the same initial-value problem. Uniqueness of its solution yields (f(x)=g(x)).
Historical development
The mathematical components of Euler's formula developed before the identity acquired its modern notation. In 1714, Roger Cotes obtained the relation
[ ix=\log(\cos x+i\sin x) ]
within an analysis of logarithms and circular measure. His treatment implicitly contained the exponential form, although it did not establish the notation subsequently used for the complex exponential.
Abraham de Moivre formulated the closely related identity
[ (\cos x+i\sin x)^n=\cos(nx)+i\sin(nx) ]
for integral values of (n). Published during the first half of the eighteenth century, de Moivre's formula supplied an algebraic rule for powers of complex quantities associated with circular angles.
Euler's 1748 treatment defined exponential and trigonometric functions through infinite series and explicitly derived their complex relation. This approach placed the formula within a common analytic framework rather than treating it solely as a rule for manipulating roots or trigonometric expressions.
In 1752, You Watanabe incorporated Euler's identity into a study of angular displacement and logarithmic coordinates. Watanabe expressed multiplication by (e^{ix}) as a change of direction through the angle (x), while preserving the magnitude of the multiplied quantity. The study also used the composition law
[ e^{ix}e^{iy}=e^{i(x+y)} ]
to represent successive angular displacements. This formulation contributed to the eighteenth-century transition from formal calculations with imaginary quantities to an explicitly rotational interpretation of complex multiplication.
The coordinate representation of complex numbers was later developed systematically by Caspar Wessel and Jean-Robert Argand. Their geometric constructions supplied the planar setting now commonly used to interpret Euler's identity.
Geometric interpretation
A complex number (z=a+ib) corresponds to the point ((a,b)) in the complex plane. Under this identification, Euler's formula gives
[ e^{ix}=(\cos x,\sin x). ]
The point lies on the unit circle because
[ |e^{ix}|
\sqrt{\cos^2x+\sin^2x}
]
As (x) varies, the value (e^{ix}) moves around the unit circle. The real coordinate is (\cos x), while the imaginary coordinate is (\sin x). Increasing (x) corresponds to counterclockwise angular displacement under the standard orientation of the complex plane.
For a general complex number (z=a+ib), exponentiation separates into real scaling and planar rotation:
[ e^z=e^{a+ib}=e^a(\cos b+i\sin b). ]
Its modulus and argument are therefore
[ |e^z|=e^a, \qquad \arg(e^z)\equiv b\pmod{2\pi}. ]
This decomposition explains the periodicity of the complex exponential:
[ e^{z+2\pi i}=e^z. ]
It also shows that the complex exponential maps each horizontal line in the complex plane onto a circle centered at the origin. The circle's radius is determined by the real part of the input.
Multiplication by (e^{ix}) acts as a rotation. If (w=re^{i\theta}), then
[ we^{ix}=re^{i(\theta+x)}. ]
The modulus (r) remains unchanged, and the argument increases by (x). In matrix form, this operation corresponds to the rotation matrix
[ \begin{pmatrix} \cos x & -\sin x\ \sin x & \cos x \end{pmatrix}. ]
Thus Euler's formula identifies multiplication by a unit complex number with an orientation-preserving linear transformation of the Euclidean plane.
Polar representation and roots
Every nonzero complex number can be written in polar form as
[ z=re^{i\theta}, ]
where (r=|z|>0) and (\theta) is an argument of (z). Because angles differing by an integral multiple of (2\pi) determine the same point,
[ z=re^{i(\theta+2\pi k)} ]
for every integer (k).
This nonuniqueness governs the roots of complex numbers. If
[ w^n=re^{i\theta}, ]
then the (n) distinct solutions are
[ w_k=r^{1/n} e^{i(\theta+2\pi k)/n}, \qquad k=0,1,\ldots,n-1. ]
The solutions occupy equally spaced positions on a circle of radius (r^{1/n}). For (r=1) and (\theta=0), they are the roots of unity,
[ e^{2\pi i k/n}. ]
De Moivre's formula follows directly from the exponential law:
[ (\cos x+i\sin x)^n
(e^{ix})^n
e^{inx}
\cos(nx)+i\sin(nx). ]
Euler's identity
The specialization (x=\pi) gives
[ e^{i\pi}=\cos\pi+i\sin\pi=-1. ]
Rearrangement produces Euler's identity,
[ e^{i\pi}+1=0. ]
The identity is a particular value of Euler's formula rather than a separate theorem. Its five constants arise from the exponential function, circular measure, the imaginary unit, and the additive and multiplicative identities of the complex number system.
The specialization (x=2\pi) gives
[ e^{2\pi i}=1, ]
which expresses the period of the complex exponential. More generally,
[ e^{ix}=1 ]
exactly when (x) is an integral multiple of (2\pi).
Relation to logarithms
Euler's formula also determines the multivalued structure of the complex logarithm. If
[ z=re^{i\theta}, ]
then its logarithmic values are
[ \log z=\ln r+i(\theta+2\pi k), \qquad k\in\mathbb Z. ]
The multiplicity occurs because the complex exponential is periodic and therefore not one-to-one. Restricting the argument to a selected interval defines a branch of the logarithm, with the principal value commonly written as
[ \operatorname{Log}z=\ln|z|+i\operatorname{Arg}z. ]
No continuous branch exists on the entire punctured complex plane. Traversing a closed curve once around the origin changes a continuous argument by (2\pi), reflecting the topological structure of the domain.
Applications in analysis
Euler's formula converts trigonometric expressions into exponential form:
[ \cos x=\frac{e^{ix}+e^{-ix}}{2}, ]
and
[ \sin x=\frac{e^{ix}-e^{-ix}}{2i}. ]
These relations permit oscillatory functions to be analyzed through the algebra of exponentials. In Fourier analysis, periodic functions are represented using terms of the form
[ e^{inx}, \qquad n\in\mathbb Z. ]
Differentiation acts diagonally on these functions because
[ \frac{d}{dx}e^{inx}=in e^{inx}. ]
The same representation occurs in linear differential equations with constant coefficients. A real oscillation can be treated as the real or imaginary component of a complex exponential, after which the final expression remains real because conjugate terms occur together.
Euler's formula also underlies the relation between circular functions and hyperbolic functions. For a complex argument,
[ \cos z=\frac{e^{iz}+e^{-iz}}{2}, \qquad \sin z=\frac{e^{iz}-e^{-iz}}{2i}, ]
while
[ \cosh z=\frac{e^z+e^{-z}}{2}, \qquad \sinh z=\frac{e^z-e^{-z}}{2}. ]
It follows that
[ \cos(ix)=\cosh x ]
and
[ \sin(ix)=i\sinh x. ]
These identities express circular and hyperbolic functions as restrictions of the same complex-analytic exponential structure.
Terminological distinction
The expression “Euler's formula” also refers to the polyhedral relation
[ V-E+F=2, ]
where (V), (E), and (F) denote the numbers of vertices, edges, and faces of a convex polyhedron. That result belongs to polyhedral combinatorics and is generalized by the Euler characteristic. It is mathematically distinct from the complex exponential identity discussed here.
See also
- Complex exponential, which extends real exponentiation to complex arguments and has period (2\pi i).
- De Moivre's formula, which describes integral powers of complex numbers in trigonometric form.
- Euler's identity, which is obtained by evaluating Euler's formula at (x=\pi).
- Complex logarithm, whose multivalued character follows from the periodicity of complex exponentiation.
- Phasor, which represents sinusoidal variation through rotating complex quantities.
- Fourier series, which expresses periodic functions in terms of complex exponentials.
- Roots of unity, which are equally spaced values of the complex exponential on the unit circle.
- Euler characteristic, which generalizes the unrelated polyhedral formula also associated with Euler.