Euler's formula

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Euler's formula or Euler's relation, attributed to Leonhard Euler, establishes the theorem, in which:

{displaystyle e^{ix}=cos x+i,operatorname {sen} x}

{displaystyle e^{-ix}=cos x-i,operatorname {sen} x}

for all real numbers x, which represents an angle in the complex plane. Here, e is the basis of natural logarithm, i It's imaginary unity, ${displaystyle operatorname {sen} x}$ and ${displaystyle cos x}$ are the trigonometric functions sinus and cosine.

History

Roger Cotes discovered in 1714 the relationship between the trigonometric functions and the logarithm,

{displaystyle ix=ln(cos x+ioperatorname {sen} x)}

and was published in his posthumous work Harmonia mensurarum (1722), 20 years before Leonhard Euler did. Euler developed the formula using the exponential function instead of the logarithm and communicated it in a letter sent to Christian Goldbach in 1741, being published and popularized in his work Introductio in analysin infinitorum in 1748. It is interesting to note that none of the discoverers saw the geometric interpretation noted above: the view of complex numbers as points in the plane arose in 1787 by the mathematician Caspar Wessel in his only report to the Royal Danish Academy.

A century later B. Peirce concluded the deduction of the formula in front of his students, saying: "Gentlemen, surely this formula is true even if it seems paradoxical to you..."

Complex power of e

Or it is usually expressed as:

{displaystyle e^{z}=e^{x+iy}=e^{x}(cos y+i,operatorname {sen} ,y)}

being $z$ complex variable defined by ${displaystyle z=x+iy}$

Demo

Note that this is not a proof based on the properties of complex numbers and the exponential function, but rather the definition of the complex exponential as the equivalent of the Taylor series over the real numbers for complex parameters.

The formula can be interpreted geometrically as a unit circumference in the complex plane, drawn by the function e^ix to vary $x$ about the real numbers. So, $x$ is the angle of a straight that connects the origin of the plane and a point on the unit circumference, with the actual positive axis, measured in the opposite direction to the needles of the clock and in radians. The formula is only valid if the breast and the cosine also have their arguments in radians.

Demostración usando las Series de Taylor

The formula of Euler illustrated in the complex plane.

Knowing that:

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

and so on. In addition to this, the functions ex, cos(x) and sin(x) (assuming x is a real number) can be expressed using its Taylor series around zero.

{displaystyle {begin{aligned}e^{x}&{}={frac {x^{0}}{0!}}+{frac {x^{1}}{1!}}+{frac {x^{2}}{2!}}+{frac {x^{3}}{3!}}+{frac {x^{4}}{4!}}+cdots \cos x&{}={frac {x^{0}}{0!}}-{frac {x^{2}}{2!}}+{frac {x^{4}}{4!}}-{frac {x^{6}}{6!}}+cdots \sin x&{}={frac {x^{1}}{1!}}-{frac {x^{3}}{3!}}+{frac {x^{5}}{5!}}-{frac {x^{7}}{7!}}+cdots end{aligned}}}

Another definition that can be given to you $e^x$ Based on Taylor's series is the following:

{displaystyle e^{x}={biggl (}sum _{k=0}^{infty }{frac {x^{k}}{k!}}{biggr)}}

also valid for:

{displaystyle cos x={biggl (}sum _{k=0}^{infty }{frac {(-1)^{k}x^{2k}}{(2k)!}}{biggr)}}

{displaystyle sin x={biggl (}sum _{k=0}^{infty }{frac {(-1)^{k}x^{2k+1}}{(2k+1)!}}{biggr)}}

We define each of these functions by the series above, replacing x by i z, where z is a real variable and i is the imaginary unit. This is possible because the radius of convergence is infinite in each series. So we find that:

{displaystyle {begin{aligned}e^{iz}&{}={frac {(iz)^{0}}{0!}}+{frac {(iz)^{1}}{1!}}+{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 \&{}={frac {z^{0}}{0!}}+i{frac {z^{1}}{1!}}-{frac {z^{2}}{2!}}-i{frac {z^{3}}{3!}}+{frac {z^{4}}{4!}}+i{frac {z^{5}}{5!}}-{frac {z^{6}}{6!}}-i{frac {z^{7}}{7!}}+{frac {z^{8}}{8!}}+cdots \&{}=left({frac {z^{0}}{0!}}-{frac {z^{2}}{2!}}+{frac {z^{4}}{4!}}-{frac {z^{6}}{6!}}+{frac {z^{8}}{8!}}-cdots right)+ileft({frac {z^{1}}{1!}}-{frac {z^{3}}{3!}}+{frac {z^{5}}{5!}}-{frac {z^{7}}{7!}}+cdots right)\&{}=cos(z)+isin(z)end{aligned}}}

Rearrangement is possible because each series is absolutely convergent. Replacing z = x as a real number results in the original identity as discovered by Euler.

Mathematical relevance

The formula provides a powerful connection between mathematical analysis and trigonometry. It is used to represent complex numbers in polar coordinates and allows you to define the logarithm for negative numbers and complex numbers.

Logarithm of a negative number

In this case, the Euler formula is evaluated in $x=pi$ obtaining the identity of Euler:

{displaystyle e^{mathrm {i} pi }=cos pi +mathrm {i} ,operatorname {sen} pi =-1}

{displaystyle e^{mathrm {i} pi }=-1,!}

Then, applying the natural logarithm we obtain:

{displaystyle mathrm {i} pi =ln(-1),!}

Logarithm of any negative number

As an extension of the above equation, the logarithm of any negative number is defined as:

{displaystyle ln(-a)=ln(a)+ln(-1)=ln(a)+mathrm {i} pi ,!}

. Where

0}" xmlns="http://www.w3.org/1998/Math/MathML">a▪0{displaystyle a vocabulary0}0}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f34a80ea013edb56e340b19550430a8b6dfd7b9" style="vertical-align: -0.338ex; width:5.491ex; height:2.176ex;"/>

In addition, the logarithm of a negative number can be defined in any base, from the natural logarithm and the base change formula.

Integration and derivation

An important property of Euler's formula is that it is the only mathematical function that remains in the same form (except for the imaginary unit) with the operations of integration and differentiation of the integral calculus, allowing it to be used to convert differential equations into equations with algebraic form, greatly simplifying those operations.

Of the rules of exponentiation

{displaystyle e^{a+b}=e^{a}cdot e^{b}}

and

{displaystyle (e^{a})^{b}=e^{acdot b}}

(valid for all couple of complex numbers $a$ and $b$ ), several trigonometric identities can be derived, as well as the formula of De Moivre.

Trigonometric functions

Euler's formula also allows us to interpret the sine and cosine functions as mere variations of the exponential function:

{displaystyle cos x={e^{ix}+e^{-ix} over 2}}

{displaystyle operatorname {sen} x={e^{ix}-e^{-ix} over 2i}}

From these equalities, it is possible to define the trigonometric functions for the complex numbers in this way:

{displaystyle operatorname {sen} z={cfrac {e^{zi}-e^{-zi}}{2i}}}

{displaystyle cos z={cfrac {e^{zi}+e^{-zi}}{2}}}

{displaystyle tan z={cfrac {e^{zi}-e^{-zi}}{e^{zi}+e^{-zi}}}cdot {cfrac {1}{i}}}

being ${displaystyle zin mathbb {C} }$ I mean, it belongs to the complex numbers set. These trigonometric functions comply with the laws of their similar applied to actual numbers. Be complex numbers $z$ and $w$ I mean, ${displaystyle zland win mathbb {C} }$ then the following equalitys are valid:

{displaystyle cos ^{2}{z}+operatorname {sen} ^{2}{z}=1}

{displaystyle cos(-z)=cos(z)}

{displaystyle operatorname {sen}(-z)=-operatorname {sen}(z)}

{displaystyle operatorname {sen}(z+w)=operatorname {sen}(z)cos(w)+operatorname {sen}(w)cos(z)}

{displaystyle cos(z+w)=cos(z)cos(w)-operatorname {sen}(w)operatorname {sen}(z)}

Differential Equations

In differential equations, expression ${displaystyle e^{ix}}$ is often used to simplify derivatives, even if the final response is a real function in which breasts or cosenos appear. The identity of Euler is an immediate consequence of the Euler formula.

Signal analysis

Signals that vary periodically are often described as a combination of sine and cosine functions, as in Fourier analysis, and these are more conveniently expressed as the real part of an exponential function with imaginary exponent, using Euler's formula.

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