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Euler's formula
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===Applications in complex number theory=== [[Image:Euler's formula.svg|thumb|right|Euler's formula {{math|1=''e<sup>iΟ</sup>'' = cos ''Ο'' + ''i'' sin ''Ο''}} illustrated in the complex plane.]] ==== Interpretation of the formula ==== This formula can be interpreted as saying that the function {{math|''e''<sup>''iΟ''</sup>}} is a [[unit complex number]], i.e., it traces out the [[unit circle]] in the [[complex plane]] as {{mvar|Ο}} ranges through the real numbers. Here {{mvar|Ο}} 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 [[radian]]s. The original proof is based on the [[Taylor series]] expansions of the [[exponential function]] {{math|''e''<sup>''z''</sup>}} (where {{mvar|z}} is a complex number) and of {{math|sin ''x''}} and {{math|cos ''x''}} for real numbers {{mvar|x}} ([[Euler's formula#Using power series|see above]]). In fact, the same proof shows that Euler's formula is even valid for all ''complex'' numbers {{mvar|x}}. A point in the [[complex plane]] can be represented by a complex number written in [[Coordinates (elementary mathematics)#Cartesian coordinates|cartesian coordinates]]. Euler's formula provides a means of conversion between cartesian coordinates and [[Polar coordinate system|polar coordinates]]. The polar form simplifies the mathematics when used in multiplication or powers of complex numbers. Any complex number {{math|1 = ''z'' = ''x'' + ''iy''}}, and its complex conjugate, {{math|1 = {{overline|''z''}} = ''x'' β ''iy''}}, can be written as <math display="block">\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}</math> where *{{math|1=''x'' = Re ''z''}} is the real part, *{{math|1=''y'' = Im ''z''}} is the imaginary part, *{{math|1=''r'' = {{abs|''z''}} = {{sqrt|''x''<sup>2</sup> + ''y''<sup>2</sup>}}}} is the [[magnitude (mathematics)|magnitude]] of {{mvar|z}} and *{{math|1=''Ο'' = arg ''z'' = [[atan2]](''y'', ''x'')}}. {{mvar|Ο}} is the [[arg (mathematics)|argument]] of {{mvar|z}}, i.e., the angle between the ''x'' axis and the vector ''z'' measured counterclockwise in [[radian]]s, which is defined [[up to]] addition of {{math|2''Ο''}}. Many texts write {{math|1=''Ο'' = tan<sup>β1</sup> ''{{sfrac|y|x}}''}} instead of {{math|1= ''Ο'' = atan2(''y'', ''x'')}}, but the first equation needs adjustment when {{math|''x'' β€ 0}}. This is because for any real {{mvar|x}} and {{mvar|y}}, not both zero, the angles of the vectors {{math|(''x'', ''y'')}} and {{math|(β''x'', β''y'')}} differ by {{pi}} radians, but have the identical value of {{math|1=tan ''Ο'' = {{sfrac|''y''|''x''}}}}. ==== 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): <math display="block">a = e^{\ln a}, </math> and that <math display="block">e^a e^b = e^{a + b}, </math> both valid for any complex numbers {{mvar|a}} and {{mvar|b}}. Therefore, one can write: <math display="block">z = \left|z\right| e^{i \varphi} = e^{\ln\left|z\right|} e^{i \varphi} = e^{\ln\left|z\right| + i \varphi}</math> for any {{math|''z'' β 0}}. Taking the logarithm of both sides shows that <math display="block">\ln z = \ln \left|z\right| + i \varphi,</math> 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 {{mvar|Ο}} is multi-valued. Finally, the other exponential law <math display="block">\left(e^a\right)^k = e^{a k},</math> which can be seen to hold for all integers {{mvar|k}}, together with Euler's formula, implies several [[trigonometric identities]], as well as [[de Moivre's formula]]. ==== Relationship to trigonometry ==== [[File:Sine Cosine Exponential qtl1.svg|thumb|Relationship between sine, cosine and exponential function]] 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 [[mathematical analysis|analysis]] and [[trigonometry]], and provides an interpretation of the sine and cosine functions as [[weighted sum]]s of the exponential function: <math display="block">\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}</math> The two equations above can be derived by adding or subtracting Euler's formulas: <math display="block">\begin{align} e^{ix} &= \cos x + i \sin x, \\ e^{-ix} &= \cos(- x) + i \sin(- x) = \cos x - i \sin x \end{align}</math> and solving for either cosine or sine. These formulas can even serve as the definition of the trigonometric functions for complex arguments {{mvar|x}}. For example, letting {{math|1=''x'' = ''iy''}}, we have: <math display="block">\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}</math> In addition <math display="block">\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}</math> 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''.<ref>{{Cite web |title=Complex Sinusoids |url=https://ccrma.stanford.edu/~jos/filters06/Complex_Sinusoids.html |access-date=2024-09-10 |website=ccrma.stanford.edu}}</ref> After the manipulations, the simplified result is still real-valued. For example: <math display="block">\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} </math> 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: <math display="block">\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}</math> This formula is used for recursive generation of {{math|cos ''nx''}} for integer values of {{mvar|n}} and arbitrary {{mvar|x}} (in radians). Considering {{math|cos ''x''}} a parameter in equation above yields recursive formula for [[Chebyshev polynomials]] of the first kind. {{see also|Phasor#Arithmetic}}
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