Trigonometric Form Of Complex Numbers

Circuitous exponential in terms of sine and cosine

Euler'due south formula, named after Leonhard Euler, is a mathematical formula in circuitous assay that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for any real number $x$ :

east^{ix}=\cos x+i\sin ten,

where $e$ is the base of operations of the natural logarithm, $i$ is the imaginary unit, and $cos$ and $sin$ are the trigonometric functions cosine and sine respectively. This complex exponential function is sometimes denoted $cis x$ ("cosine plus i sine"). The formula is still valid if $x$ is a complex number, and and then some authors refer to the more general complex version every bit Euler's formula.^[ane]

Euler'southward formula is ubiquitous in mathematics, physics, and engineering. The physicist Richard Feynman chosen the equation "our jewel" and "the nearly remarkable formula in mathematics".^[ii]

When $ten = π$ , Euler's formula may be rewritten every bit $eastward iπ + 1 = 0$ , which is known every bit Euler's identity.

History [edit]

In 1714, the English mathematician Roger Cotes presented a geometrical statement that can be interpreted (later on correcting a misplaced gene of ${\sqrt {-1}}$ ) as:^[3] ^[4] ^[v]

ix=\ln(\cos x+i\sin x).

Exponentiating this equation yields Euler'due south formula. Note that the logarithmic statement is not universally right for complex numbers, since a complex logarithm can have infinitely many values, differing by multiples of $2 πi$ .

Around 1740 Leonhard Euler turned his attention to the exponential part and derived the equation named after him by comparing the serial expansions of the exponential and trigonometric expressions.^[6] ^[4] The formula was first published in 1748 in his foundational work Introductio in analysin infinitorum.^[7]

Johann Bernoulli had found that^[8]

{\frac {1}{1+10^{ii}}}={\frac {one}{ii}}\left({\frac {1}{1-nine}}+{\frac {1}{1+ix}}\correct).

And since

\int {\frac {dx}{1+ax}}={\frac {1}{a}}\ln(ane+ax)+C,

the above equation tells u.s. something nigh circuitous logarithms by relating natural logarithms to imaginary (complex) numbers. Bernoulli, withal, did not evaluate the integral.

Bernoulli's correspondence with Euler (who also knew the above equation) shows that Bernoulli did not fully empathise circuitous logarithms. Euler also suggested that circuitous logarithms tin accept infinitely many values.

The view of complex numbers equally points in the complex plane was described most l years afterward by Caspar Wessel.

Definitions of circuitous exponentiation [edit]

The exponential function $e x$ for real values of $10$ may be defined in a few different equivalent ways (come across Characterizations of the exponential part). Several of these methods may exist directly extended to requite definitions of $e z$ for complex values of $z$ simply by substituting $z$ in place of $x$ and using the circuitous algebraic operations. In particular we may use any of the three post-obit definitions, which are equivalent. From a more advanced perspective, each of these definitions may be interpreted equally giving the unique analytic continuation of $e x$ to the complex airplane.

Differential equation definition [edit]

The exponential part $z\mapsto east^{z}$ is the unique differentiable function of a complex variable for which the derivative equals the office

{\frac {de^{z}}{dz}}=east^{z}

and

e^{0}=one.

Power series definition [edit]

For complex $z$

eastward^{z}=1+{\frac {z}{1!}}+{\frac {z^{2}}{ii!}}+{\frac {z^{3}}{3!}}+\cdots =\sum _{n=0}^{\infty }{\frac {z^{n}}{northward!}}.

Using the ratio test, it is possible to show that this ability serial has an infinite radius of convergence and so defines $east z$ for all complex $z$ .

Limit definition [edit]

For complex $z$

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

Here, $north$ is restricted to positive integers, and so there is no question well-nigh what the power with exponent $north$ means.

Proofs [edit]

Various proofs of the formula are possible.

Using differentiation [edit]

This proof shows that the quotient of the trigonometric and exponential expressions is the abiding role ane, then they must be equal (the exponential function is never zero,^[9] so this is permitted).^[10]

Consider the office $f (θ)$

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

for existent $θ$ . Differentiating gives past the product rule

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 $f (θ) = ane$ for all existent $θ$ , and thus

eastward^{i\theta }=\cos \theta +i\sin \theta .

Using ability serial [edit]

Here is a proof of Euler's formula using power-serial expansions, as well every bit basic facts nearly the powers of $i$ :^[11]

{\brainstorm{aligned}i^{0}&=1,&i^{i}&=i,&i^{2}&=-ane,&i^{three}&=-i,\\i^{four}&=one,&i^{five}&=i,&i^{vi}&=-i,&i^{vii}&=-i\\&\vdots &&\vdots &&\vdots &&\vdots \end{aligned}}

Using now the power-series definition from above, nosotros come across that for real values of $x$

{\begin{aligned}east^{ix}&=1+ix+{\frac {(ix)^{ii}}{2!}}+{\frac {(nine)^{3}}{3!}}+{\frac {(nine)^{4}}{4!}}+{\frac {(ix)^{5}}{5!}}+{\frac {(ix)^{6}}{half-dozen!}}+{\frac {(ix)^{vii}}{vii!}}+{\frac {(ix)^{eight}}{8!}}+\cdots \\[8pt]&=1+9-{\frac {x^{2}}{ii!}}-{\frac {nine^{3}}{3!}}+{\frac {x^{4}}{4!}}+{\frac {nine^{v}}{5!}}-{\frac {x^{6}}{6!}}-{\frac {ix^{7}}{7!}}+{\frac {ten^{8}}{8!}}+\cdots \\[8pt]&=\left(1-{\frac {x^{ii}}{2!}}+{\frac {x^{4}}{four!}}-{\frac {x^{half-dozen}}{half-dozen!}}+{\frac {ten^{8}}{eight!}}-\cdots \correct)+i\left(x-{\frac {x^{iii}}{three!}}+{\frac {x^{v}}{5!}}-{\frac {x^{7}}{7!}}+\cdots \correct)\\[8pt]&=\cos x+i\sin x,\end{aligned}}

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

Using polar coordinates [edit]

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

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

No assumptions are being made about $r$ and $θ$ ; they volition be determined in the course of the proof. From any of the definitions of the exponential function it tin be shown that the derivative of $e ix$ is $ie ix$ . Therefore, differentiating both sides gives

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 $e nine$ and equating existent and imaginary parts in this formula gives $dr / dx = 0$ and $dθ / dx = 1$ . Thus, $r$ is a constant, and $θ$ is $ten + C$ for some constant $C$ . The initial values $r (0) = 1$ and $θ (0) = 0$ come up from $east 0 i = 1$ , giving $r = 1$ and $θ = ten$ . This proves the formula

east^{nine}=1(\cos x+i\sin x)=\cos ten+i\sin x.

Applications [edit]

Applications in complex number theory [edit]

Euler's formula $e iφ = cos φ + i sin φ$ illustrated in the complex plane.

Estimation of the formula [edit]

This formula tin exist interpreted as maxim that the function $e iφ$ is a unit complex number, i.eastward., it traces out the unit of measurement circle in the complex plane as $φ$ ranges through the real numbers. Hither $φ$ is the angle that a line connecting the origin with a indicate on the unit circumvolve makes with the positive existent centrality, measured counterclockwise and in radians.

The original proof is based on the Taylor series expansions of the exponential office $e z$ (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 $x$ .

A point in the circuitous plane can exist represented by a circuitous number written in cartesian coordinates. Euler'south formula provides a ways of conversion between cartesian coordinates and polar coordinates. The polar form simplifies the mathematics when used in multiplication or powers of complex numbers. Any circuitous number $z = x + iy$ , and its circuitous conjugate, $z = x - iy$ , tin exist written as

{\begin{aligned}z&=x+iy=|z|(\cos \varphi +i\sin \varphi )=re^{i\varphi },\\{\bar {z}}&=ten-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,
$r = | z | = \sqrt ten two + y ii$ is the magnitude of $z$ and
$φ = arg z = atan2(y, 10)$ .

$φ$ is the argument of $z$ , i.due east., the bending betwixt the x axis and the vector z measured counterclockwise in radians, which is defined up to add-on of $two π$ . Many texts write $φ = tan -one y / x$ instead of $φ = atan2(y, ten)$ , only the first equation needs adjustment when $x \leq 0$ . This is because for any existent $10$ and $y$ , non both zero, the angles of the vectors $(x, y)$ and $(- x, - y)$ differ by $π$ radians, just have the identical value of $tan φ = y / x$ .

Use of the formula to define the logarithm of circuitous numbers [edit]

Now, taking this derived formula, we can use Euler's formula to define the logarithm of a circuitous number. To practice this, nosotros also use the definition of the logarithm (as the changed operator of exponentiation):

a=e^{\ln a},

and that

e^{a}due east^{b}=east^{a+b},

both valid for whatever complex numbers $a$ and $b$ . Therefore, 1 tin write:

z=\left|z\right|eastward^{i\varphi }=eastward^{\ln \left|z\right|}e^{i\varphi }=due east^{\ln \left|z\right|+i\varphi }

for whatever $z \neq 0$ . Taking the logarithm of both sides shows that

\ln z=\ln \left|z\right|+i\varphi ,

and in fact, this can exist 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

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

which can exist seen to agree for all integers $thousand$ , together with Euler'due south formula, implies several trigonometric identities, too as de Moivre'southward formula.

Relationship to trigonometry [edit]

Relationship between sine, cosine and exponential role

Euler's formula, the definitions of the trigonometric functions and the standard identities for exponentials are sufficient to easily derive almost trigonometric identities. It provides a powerful connectedness between analysis and trigonometry, and provides an interpretation of the sine and cosine functions equally weighted sums of the exponential function:

{\begin{aligned}\cos ten&=\operatorname {Re} \left(eastward^{9}\correct)={\frac {e^{9}+e^{-ix}}{2}},\\\sin ten&=\operatorname {Im} \left(e^{9}\right)={\frac {east^{nine}-due east^{-ix}}{2i}}.\stop{aligned}}

The 2 equations above can exist derived by adding or subtracting Euler's formulas:

{\begin{aligned}e^{ix}&=\cos x+i\sin x,\\due east^{-ix}&=\cos(-x)+i\sin(-x)=\cos x-i\sin x\cease{aligned}}

and solving for either cosine or sine.

These formulas can fifty-fifty serve equally the definition of the trigonometric functions for complex arguments $x$ . For instance, letting $ten = iy$ , we have:

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

Complex exponentials can simplify trigonometry, considering they are easier to dispense 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:

{\brainstorm{aligned}\cos x\cos y&={\frac {eastward^{9}+e^{-ix}}{2}}\cdot {\frac {e^{iy}+eastward^{-iy}}{2}}\\&={\frac {1}{ii}}\cdot {\frac {e^{i(10+y)}+e^{i(10-y)}+due east^{i(-x+y)}+e^{i(-x-y)}}{2}}\\&={\frac {1}{2}}{\bigg (}\underbrace {\frac {e^{i(x+y)}+eastward^{-i(x+y)}}{2}} _{\cos(ten+y)}+\underbrace {\frac {e^{i(x-y)}+e^{-i(ten-y)}}{2}} _{\cos(x-y)}{\bigg )}.\end{aligned}}

Another technique is to represent the sinusoids in terms of the existent office of a circuitous expression and perform the manipulations on the complex expression. For example:

{\brainstorm{aligned}\cos nx&=\operatorname {Re} \left(e^{inx}\right)\\&=\operatorname {Re} \left(e^{i(n-1)x}\cdot e^{ix}\correct)\\&=\operatorname {Re} {\Large (}e^{i(n-1)10}\cdot {\big (}\underbrace {east^{nine}+east^{-9}} _{2\cos x}-e^{-9}{\big )}{\Large )}\\&=\operatorname {Re} \left(due east^{i(north-1)10}\cdot 2\cos x-e^{i(n-two)x}\right)\\&=\cos[(n-1)x]\cdot [2\cos 10]-\cos[(n-2)x].\end{aligned}}

This formula is used for recursive generation of $cos nx$ for integer values of $north$ and arbitrary $x$ (in radians).

Topological interpretation [edit]

In the language of topology, Euler's formula states that the imaginary exponential role $t\mapsto eastward^{it}$ 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 beneath:

Other applications [edit]

In differential equations, the function $due east ix$ is ofttimes used to simplify solutions, even if the final reply is a real office involving sine and cosine. The reason for this is that the exponential office is the eigenfunction of the operation of differentiation.

In electrical applied science, signal processing, and similar fields, signals that vary periodically over time are often described every bit a combination of sinusoidal functions (come across Fourier analysis), and these are more than conveniently expressed every bit the sum of exponential functions with imaginary exponents, using Euler'due south formula. Besides, phasor analysis of circuits can include Euler'due south formula to represent the impedance of a capacitor or an inductor.

In the 4-dimensional space of quaternions, at that place is a sphere of imaginary units. For any point $r$ on this sphere, and $x$ a real number, Euler'south formula applies:

\exp xr=\cos ten+r\sin x,

and the chemical element is chosen a versor in quaternions. The set of all versors forms a three-sphere in the 4-space.

Encounter also [edit]

Complex number
Euler's identity
Integration using Euler's formula
History of Lorentz transformations § Euler's gap
Listing of things named afterward Leonhard Euler

References [edit]

^ Moskowitz, Martin A. (2002). A Grade in Circuitous Assay in One Variable. Globe Scientific Publishing Co. p. vii. ISBN981-02-4780-X.
^ Feynman, Richard P. (1977). The Feynman Lectures on Physics, vol. I. Addison-Wesley. p. 22-10. ISBN0-201-02010-vi.
^
Cotes wrote: "Nam si quadrantis circuli quilibet arcus, radio CE descriptus, sinun habeat CX sinumque complementi advertisement quadrantem XE ; sumendo radium CE pro Modulo, arcus erit rationis inter $EX+XC{\sqrt {-1}}$ & CE mensura ducta in ${\sqrt {-1}}$ ." (Thus if any arc of a quadrant of a circumvolve, described by the radius CE, has sinus CX and sinus of the complement to the quadrant XE ; taking the radius CE as modulus, the arc will be the measure of the ratio between $EX+90{\sqrt {-1}}$ $EX+XC{\sqrt {-1}}$ & CE multiplied past ${\sqrt {-1}}$ ${\sqrt {-1}}$ .) That is, consider a circle having center Eastward (at the origin of the (x,y) plane) and radius CE. Consider an angle θ with its vertex at Eastward having the positive x-axis equally one side and a radius CE as the other side. The perpendicular from the point C on the circumvolve to the x-axis is the "sinus" CX ; the line between the circle's center E and the bespeak 10 at the foot of the perpendicular is XE, which is the "sinus of the complement to the quadrant" or "cosinus". The ratio between $EX+Xc{\sqrt {-1}}$ $EX+XC{\sqrt {-1}}$ and CE is thus $\cos \theta +{\sqrt {-1}}\sin \theta \$ $\cos \theta +{\sqrt {-1}}\sin \theta \$ . In Cotes' terminology, the "measure" of a quantity is its natural logarithm, and the "modulus" is a conversion factor that transforms a measure of angle into round arc length (here, the modulus is the radius (CE) of the circle). According to Cotes, the product of the modulus and the measure (logarithm) of the ratio, when multiplied by ${\sqrt {-1}}$ ${\sqrt {-1}}$ , equals the length of the round arc subtended past θ, which for whatever angle measured in radians is CE • θ. Thus, ${\sqrt {-1}}CE\ln {\left(\cos \theta +{\sqrt {-1}}\sin \theta \correct)\ }=(CE)\theta$ ${\sqrt {-1}}CE\ln {\left(\cos \theta +{\sqrt {-1}}\sin \theta \right)\ }=(CE)\theta$ . This equation has the incorrect sign: the factor of ${\sqrt {-1}}$ ${\sqrt {-1}}$ should exist on the correct side of the equation, non the left side. If this change is made, then, after dividing both sides by CE and exponentiating both sides, the issue is: $\cos \theta +{\sqrt {-one}}\sin \theta =e^{{\sqrt {-1}}\theta }$ $\cos \theta +{\sqrt {-1}}\sin \theta =e^{{\sqrt {-1}}\theta }$ , which is Euler'south formula.
Run into:
- Roger Cotes (1714) "Logometria," Philosophical Transactions of the Royal Society of London, 29 (338) : 5-45 ; see especially folio 32. Available on-line at: Hathi Trust
- Roger Cotes with Robert Smith, ed., Harmonia mensurarum … (Cambridge, England: 1722), chapter: "Logometria", p. 28.
^ ^a ^b John Stillwell (2002). Mathematics and Its History. Springer. ISBN9781441960528.
^ Sandifer, C. Edward (2007), Euler's Greatest Hits, Mathematical Clan of America ISBN 978-0-88385-563-8
^ Leonard Euler (1748) Chapter viii: On transcending quantities arising from the circle of Introduction to the Assay of the Infinite, page 214, department 138 (translation past Ian Bruce, pdf link from 17 century maths).
^ Conway & Guy, pp. 254–255
^ Bernoulli, Johann (1702). "Solution d'un problème concernant le calcul intégral, avec quelques abrégés par rapport à ce calcul" [Solution of a trouble in integral calculus with some notes relating to this calculation]. Mémoires de fifty'Académie Royale des Sciences de Paris. 1702: 289–297.
^ Apostol, Tom (1974). Mathematical Analysis. Pearson. p. 20. ISBN978-0201002881. Theorem 1.42
^ user02138 (https://math.stackexchange.com/users/2720/user02138), How to bear witness Euler's formula: $eastward^{i\varphi}=\cos(\varphi) +i\sin(\varphi)$?, URL (version: 2018-06-25): https://math.stackexchange.com/q/8612
^ Ricardo, Henry J. (23 March 2016). A Modern Introduction to Differential Equations. p. 428. ISBN9780123859136.
^ Strang, Gilbert (1991). Calculus. Wellesley-Cambridge. p. 389. ISBN0-9614088-2-0. Second proof on folio.

Farther reading [edit]

Nahin, Paul J. (2006). Dr. Euler'southward Fabulous Formula: Cures Many Mathematical Ills . Princeton Academy Printing. ISBN978-0-691-11822-2.
Wilson, Robin (2018). Euler's Pioneering Equation: The Near Beautiful Theorem in Mathematics. Oxford: Oxford Academy Press. ISBN978-0-nineteen-879492-9. MR 3791469.

External links [edit]

Elements of Algebra