Differential equation definition

The exponential function ${\displaystyle f(z)=e^{z}}$ is the unique differentiable function of a complex variable for which the derivative equals the function

$${\displaystyle {\frac {df}{dz}}=f}$$

and

$${\displaystyle f(0)=1.}$$

Power series definition

For complex z

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

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

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

Here, n is restricted to positive integers, so there is no question about what the power with exponent n means.

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,^[9] so this is permitted).^[10]

Consider the function f(θ)

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

for real θ. Differentiating gives by the product rule

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

Thus, f(θ) is a constant. Since f(0) = 1, then f(θ) = 1 for all real θ, and thus

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

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:^[11]

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

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

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

where in the last step we recognize the two terms are the Maclaurin series for cos x and sin x. The rearrangement of terms is justified because each series is absolutely convergent.

Using polar coordinates

Another proof^[12] is based on the fact that all complex numbers can be expressed in polar coordinates. Therefore, for some r and θ depending on x,

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

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 e^ix is ie^ix. Therefore, differentiating both sides gives

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

Substituting r(cos θ + i sin θ) for e^ix 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. The initial values r(0) = 1 and θ(0) = 0 come from e⁰ⁱ = 1, giving r = 1 and θ = x. This proves the formula

$${\displaystyle e^{i\theta }=1(\cos \theta +i\!\sin \theta )=\cos \theta +i\!\sin \theta .}$$

Applications in complex number theory

Euler's formula 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 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 e^z (where z is a complex number) and of sin x and 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 z = x + iy, and its complex conjugate, z = x − iy, can be written as

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

where

• x = Re z is the real part,
• y = Im z is the imaginary part,
• r = |z| = √x² + y² is the magnitude of z and
• 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 2π. Many texts write φ = tan⁻¹ y/x instead of φ = atan2(y, x), but the first equation needs adjustment when x ≤ 0. This is because for any real x and y, not both zero, the angles of the vectors (x, y) and (−x, −y) differ by π radians, but have the identical value of 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):

$${\displaystyle a=e^{\ln a},}$$

and that

$${\displaystyle e^{a}e^{b}=e^{a+b},}$$

both valid for any complex numbers a and b. Therefore, one can write:

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

for any z ≠ 0. Taking the logarithm of both sides shows that

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

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

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

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:

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

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

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

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 x = iy, we have:

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

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:

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

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:

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

This formula is used for recursive generation of 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 ${\displaystyle t\mapsto e^{it}}$ is a (surjective) morphism of topological groups from the real line ${\displaystyle \mathbb {R} }$ to the unit circle ${\displaystyle \mathbb {S} ^{1}}$. In fact, this exhibits ${\displaystyle \mathbb {R} }$ as a covering space of ${\displaystyle \mathbb {S} ^{1}}$. Similarly, Euler's identity says that the kernel of this map is ${\displaystyle \tau \mathbb {Z} }$, where ${\displaystyle \tau =2\pi }$. These observations may be combined and summarized in the commutative diagram below:

Euler's formula and identity combined in diagrammatic form

Other applications

In differential equations, the function 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:

$${\displaystyle \exp xr=\cos x+r\sin x,}$$

and the element is called a versor in quaternions. The set of all versors forms a 3-sphere in the 4-space.