Applications in complex number theory Interpretation of the formula This formula can be interpreted as saying that the function is a
unit complex number, i.e., it traces out the
unit circle in the
complex plane as ranges through the real numbers. Here 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 (where is a complex number) and of and for real numbers (
see above). In fact, the same proof shows that Euler's formula is even valid for all
complex numbers . 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 , and its complex conjugate, , can be written as \begin{align} z &= x + iy = |z| (\cos \varphi + i\sin \varphi) = r e^{i \varphi}, \\ \bar{z} &= x - iy = |z| (\cos \varphi - i\sin \varphi) = r e^{-i \varphi}, \end{align} where • is the real part, • is the imaginary part, • \textstyle r = |z| = \sqrt{x^2 + y^2} is the
magnitude of and • . is the
argument of , i.e., the angle between the
x axis and the vector
z measured counterclockwise in
radians, which is defined
up to addition of . Many texts write instead of , but the first equation needs adjustment when . This is because for any real and , not both zero, the angles of the vectors and differ by radians, but have the identical value of .
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): a = e^{\ln a}, and that e^a e^b = e^{a + b}, both valid for any complex numbers and . Therefore, one can write: z = \left|z\right| e^{i \varphi} = e^{\ln\left|z\right|} e^{i \varphi} = e^{\ln\left|z\right| + i \varphi} for any . Taking the logarithm of both sides shows that \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 is multi-valued. Finally, the other exponential law \left(e^a\right)^k = e^{a k}, which can be seen to hold for all integers , together with Euler's formula, implies several
trigonometric identities, as well as
de Moivre's formula.
Relationship to trigonometry 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: \begin{align} \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{align} The two equations above can be derived by adding or subtracting Euler's formulas: \begin{align} e^{ix} &= \cos x + i \sin x, \\ e^{-ix} &= \cos(- x) + i \sin(- x) = \cos x - i \sin x \end{align} and solving for either cosine or sine. These formulas can even serve as the definition of the trigonometric functions for complex arguments . For example, letting , we have: \begin{align} \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{align} In addition \begin{align} \cosh ix &= \frac{e^{ix} + e^{-ix}}{2} = \cos x, \\ \sinh ix &= \frac{e^{ix} - e^{-ix}}{2} = i\sin x. \end{align} Complex exponentials can simplify trigonometry, because they are mathematically easier to manipulate than their sine and cosine components. One technique is simply to convert sines and cosines into equivalent expressions in terms of exponentials sometimes called
complex sinusoids. After the manipulations, the simplified result is still real-valued. For example: \begin{align} \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{align} Another technique is to represent sines and cosines in terms of the
real part of a complex expression and perform the manipulations on the complex expression. For example: \begin{align} \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{align} This formula is used for recursive generation of for integer values of and arbitrary (in radians). Considering a parameter in equation above yields recursive formula for
Chebyshev polynomials of the first kind.
Topological interpretation 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:
Other applications In
differential equations, the function 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 on this sphere, and 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. ==Other special cases==