Euler's formula

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This article is about Euler's formula in complex analysis. For other meanings, see Euler function.

Image:Euler's formula.png

Euler's formula, named after Leonhard Euler (pronounced oiler), is a mathematical formula in the subfield of 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,

<math>e^{ix} = \cos x + i\sin x \!</math>

where

<math>e</math> is the base of the natural logarithm

<math>i</math> is the imaginary unit

<math>\sin</math> and <math>\cos</math> are trigonometric functions.

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

Contents 1 History 2 Applications in complex number theory 3 Relationship to trigonometry 4 Other applications 5 Proofs 5.1 Using Taylor series 5.2 Using calculus 6 References 7 External links 8 See also

History

Euler's formula was proved (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. For any complex number z, we have the following relationships:

<math> z = x + iy = Ae^{i\phi} \!</math>

where

<math> x = \mathrm{Re}\{z\} \!</math>

<math> y = \mathrm{Im}\{z\} \!</math>

<math> A = |z| \!</math>

and

<math> \phi = \angle z \!</math>

The formula also provides a definition for the natural logarithm of complex arguments:

<math>\ln(z) = \ln(Ae^{i\phi}) = \ln(A) + \ln(e^{i\phi}) = \ln(A) + i\phi \!</math>

for z≠0. Therefore,

<math> \mathrm{Re} \{ \ln(z) \} = \ln(A) = \ln(|z|) \!</math>

and

<math> \mathrm{Im} \{ \ln(z) \} = \phi = \angle z \!</math>

By using the exponential laws

<math>e^{a + b} = e^a \cdot e^{b}</math>

and

<math>(e^a)^b = e^{a b} \,</math>

(which are valid for all complex numbers a and b), one can also readily derive 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:

<math>\cos x = {e^{ix} + e^{-ix} \over 2}</math>

<math>\sin x = {e^{ix} - e^{-ix} \over 2i}</math>

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

<math>e^{ix} = \cos x + i \sin x \;</math>

<math>e^{-ix} = \cos x - i \sin x \;</math>

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:

<math> \cos(iy) = {e^{-y} + e^{y} \over 2} = \cosh(y) </math>

<math> \sin(iy) = {e^{-y} - e^{y} \over 2i} = i \sinh(y). </math>

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:

<math>i^0=1 \,</math>

<math>i^1=i \,</math>

<math>i^2=-1 \,</math>

<math>i^3=-i \,</math>

<math>i^4=1 \,</math>

<math>i^5=i \,</math>

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

<math> e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots </math>

<math> \cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots

</math>

<math> \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots

</math>

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

<math>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</math>

<math>= 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</math>

<math>= \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) </math>

<math>= \cos (z) + i\sin (z) \,</math>

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 complex number <math>z</math> such that

<math>z=\cos x + i\sin x \,</math> (ignoring the modulus term, as this cancels later)

Differentiating <math>z</math> with respect to <math>x</math>:

<math>\frac{dz}{dx}=-\sin x + i\cos x</math>

Using the fact that i² = -1:

<math>\frac{dz}{dx}=i^2\sin x + i\cos x=i(\cos x + i\sin x)=iz</math>

Separating variables and integrating both sides:

<math>\int\frac{1}{z}\,dz=\int i\,dx</math>

<math>\ln z=ix + C\,</math>

where

<math>C</math> is the constant of integration.

To finish the proof we have to argue that <math>C</math> is zero. This is easily done by substituting <math>x=0</math>.

<math>\ln z = C\,</math>

But <math>z</math> is just equal to:

<math>z = \cos x + i\sin x = \cos 0 + i \sin 0 = 1 \,</math>

thus

<math>\ln 1 = C \,</math>

<math>C = 0 \,</math>

So now we just exponentiate

<math>\ln z = ix \,</math>

<math>e^{\ln z} = e^{ix} \,</math>

<math>z = e^{ix} \,</math>

<math>e^{ix} = \cos x + i\sin x \,</math>

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

See also

• Leonhard Euler
• Euler's identity
• Complex number
• Exponential function
• Trigonometrybg:Формула на Ойлер

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