This article is about Euler's formula in complex analysis. For Euler's formula in graph theory see planar graph. See also topics named after Euler.

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.)

Euler's formula states that, for any real number x,

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

where

$e$ is the base of the natural logarithm

$i$ is the imaginary unit

$\backslash sin$ and $\backslash cos$ are trigonometric functions.

Richard Feynman called Euler's formula "our jewel" and "the most remarkable formula in mathematics" (Feynman, p. 22-10).

History

---

Euler's formula was proven (in an obscured form) for the first time by Roger Cotes in 1714, then rediscovered and popularized by Euler in 1748. Neither of these men saw the geometrical interpretation of the formula: the view of complex numbers as points in the plane arose only some 50 years later (see Caspar Wessel).

Applications in complex number theory

---

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 formula is valid only if sin and cos take their arguments in radians rather than in degrees.

The 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 x.

Euler's formula can be used to represent complex numbers in polar coordinates. Any complex number z=x+iy can be written as

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

where

$x\; =\; \backslash mathrm\{Re\}\backslash \{z\backslash \}\; \backslash ,$

$y\; =\; \backslash mathrm\{Im\}\backslash \{z\backslash \}\; \backslash ,$

$|z|$ is the magnitude of z

and $\backslash phi$ is the argument of z— 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 facts that

$a\; =\; e^\{ln\; (a)\}\backslash ,$

and

$e^a\; e^\{b\}\; =\; e^\{a\; +\; b\}\backslash ,$

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 $z\backslash ne\; 0$. Taking the logarithm of both sides shows that:

$\backslash ln\; z=\; \backslash ln\; |z|\; +\; i\; \backslash phi.\backslash ,$

and in fact this can be used as the defintion for the complex logarithm. The logarithm of a complex number is thus a multi-valued function, due to the fact that $\backslash phi$ is multi-valued.

Finally, the other exponential law

$(e^a)^k\; =\; e^\{a\; k\},\; \backslash ,$

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:

$\backslash cos\; x\; =\; \{e^\{ix\}\; +\; e^\{-ix\}\; \backslash over\; 2\}$

$\backslash sin\; x\; =\; \{e^\{ix\}\; -\; e^\{-ix\}\; \backslash over\; 2i\}$

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

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

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

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:

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

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

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. 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.

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:

$i^0=1\; \backslash ,$

$i^1=i\; \backslash ,$

$i^2=-1\; \backslash ,$

$i^3=-i\; \backslash ,$

$i^4=1\; \backslash ,$

$i^5=i\; \backslash ,$

and so on. The functions e^x, cos(x) and sin(x) (assuming x is real) can be written as:

$e^x\; =\; 1\; +\; x\; +\; \backslash frac\{x^2\}\{2!\}\; +\; \backslash frac\{x^3\}\{3!\}\; +\; \backslash cdots$

$\backslash cos\; x\; =\; 1\; -\; \backslash frac\{x^2\}\{2!\}\; +\; \backslash frac\{x^4\}\{4!\}\; -\; \backslash frac\{x^6\}\{6!\}\; +\; \backslash cdots$

$\backslash sin\; x\; =\; x\; -\; \backslash frac\{x^3\}\{3!\}\; +\; \backslash frac\{x^5\}\{5!\}\; -\; \backslash frac\{x^7\}\{7!\}\; +\; \backslash cdots$

and for complex z we define each of these function by the above series, replacing x with iz. This is possible because the radius of convergence of each series is infinite. We then find that

$e^\{iz\}\; =\; 1\; +\; iz\; +\; \backslash frac\{(iz)^2\}\{2!\}\; +\; \backslash frac\{(iz)^3\}\{3!\}\; +\; \backslash frac\{(iz)^4\}\{4!\}\; +\; \backslash frac\{(iz)^5\}\{5!\}\; +\; \backslash frac\{(iz)^6\}\{6!\}\; +\; \backslash frac\{(iz)^7\}\{7!\}\; +\; \backslash frac\{(iz)^8\}\{8!\}\; +\; \backslash cdots$

$=\; 1\; +\; iz\; -\; \backslash frac\{z^2\}\{2!\}\; -\; \backslash frac\{iz^3\}\{3!\}\; +\; \backslash frac\{z^4\}\{4!\}\; +\; \backslash frac\{iz^5\}\{5!\}\; -\; \backslash frac\{z^6\}\{6!\}\; -\; \backslash frac\{iz^7\}\{7!\}\; +\; \backslash frac\{z^8\}\{8!\}\; +\; \backslash cdots$

$=\; \backslash left(\; 1\; -\; \backslash frac\{z^2\}\{2!\}\; +\; \backslash frac\{z^4\}\{4!\}\; -\; \backslash frac\{z^6\}\{6!\}\; +\; \backslash frac\{z^8\}\{8!\}\; -\; \backslash cdots\; \backslash right)\; +\; i\backslash left(\; z\; -\; \backslash frac\{z^3\}\{3!\}\; +\; \backslash frac\{z^5\}\{5!\}\; -\; \backslash frac\{z^7\}\{7!\}\; +\; \backslash cdots\; \backslash right)$

$=\; \backslash cos\; (z)\; +\; i\backslash sin\; (z)\; \backslash ,$

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.

Q.E.D.

Using calculus

Define the function $f$ by

$f(x)\; =\; \backslash frac\{\backslash cos\; x+i\backslash sin\; x\}\{e^\{ix\}\}.$

This is allowed since the equation

$e^\{ix\}\backslash cdot\; e^\{-ix\}=e^0=1$

implies that $e^\{ix\}$ is never zero.

The derivative of $f$ is, according to the quotient rule:

& = & \displaystyle\frac{-\sin x\cdot e^{ix}-i^2\sin x\cdot e^{ix}}{(e^{ix})^2} \\^* & = & 0.\end{matrix} Therefore, $f$ must be a constant function, and from

$f(0)\; =\; \backslash frac\{\backslash cos\; 0+i\backslash sin\; 0\}\{e^0\}\; =\; 1$

we see that this constant must be 1. Hence, the numerator and denominator in the definition of $f$ must be equal for all $x$, but this is just Euler's formula.

See also

---

• Leonhard Euler
• Euler's identity
• Complex number
• Exponential function
• Trigonometry

References

---

• Feynman, Richard P., The Feynman Lectures on Physics, vol. I Addison-Wesley (1977), ISBN 0201020106, ISBN 02010211161

External links

---

• Euler and his beautiful and extraordinary formula by Antonio Gutierrez from Geometry Step by Step from the Land of the Incas.
• Euler's Formula - Puzzle: 55 pieces in a six star style of piece by Antonio Gutierrez from Geometry Step by Step from the Land of the Incas.
• Detailed Proof of Euler's Relation by Craig Lewis.
• Proof of Euler's Formula by Julius O. Smith III
• Euler's Formula and Fermat's Last Theorem

Complex analysis | Mathematical theorems

Формула на Ойлер | Eulers formel | Eulersche Formel | Fórmula de Euler | Formule d'Euler | Formula di Eulero | נוסחת אוילר | オイラーの公式 | 오일러 공식 | Formule van Euler | Wzór Eulera | Формулы Эйлера | Eulerin kaava | Eulers formel | สูตรของออยเลอร์ | Công thức Euler