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# Euler's formula

 Definition/Summary Euler's formula, $e^{ix}\ =\ \cos x\ +\ i \sin x$, enables the trigonometric functions to be defined without reference to geometry.

 Equations $$e^{ix}\ =\ \cos x\ +\ i \sin x$$ and so cos and sin may be defined: $$\cos x\ =\ \frac{1}{2}\left(e^{ix}\ +\ e^{-ix}\right)$$ and $$\sin x\ =\ \frac{1}{2i}\left(e^{ix}\ -\ e^{-ix}\right)$$ or: $$\cos x\ =\ 1\ -\ \frac{x^2}{2} +\ \frac{x^4}{24} -\ \frac{x^6}{720}\ \dots\ = \sum_{n\,=\,0}^{\infty}\frac{(-x)^{2n}}{(2n)!}$$ $$\sin x\ =\ x\ -\ \frac{x^3}{6} +\ \frac{x^5}{120} -\ \frac{x^7}{5040}\ \dots\ = \sum_{n\,=\,0}^{\infty}\frac{(-x)^{2n+1}}{(2n+1)!}$$

 Scientists Abraham de Moivre (1667-1754) Roger Cotes (1682-1716) Leonhard Euler (1707-1783)

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 Breakdown Mathematics > Calculus/Analysis >> Functions

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 Extended explanation Proof of Euler's formula, starting from the trignonometric definitions of cos and sin: Using the chain rule: $$\frac{d}{dx}\left(e^{-ix}\,(cosx\ +\ i sinx)\right)$$ $$=\ e^{ix}\,(-i cosx\ +\ sinx\ -\ sinx\ +\ i cosx)$$ $$=\ 0$$ and so $e^{-ix}\,(cosx\ +\ i sinx)$ is a constant. Setting x = 0 we find that this constant must be 1. and so $$cosx\ +\ i sinx\ =\ e^{ix}$$ History: Euler's formula was discovered by Cotes. de Moivre's formula, $(cosx\ +\ i sinx)^n$ = $cos(nx)\ +\ i sin(nx)$, is an obvious consequence of Euler's formula, but was discovered earlier.

Commentary

 FPinget @ 06:50 AM Jan10-13 Euler's equation eix = cosx + isinx is used to describe many physical events, among them the phase relationship between current and voltage in simple AC Resistor-Inductor-Capacitor type of circuits. Fernando Pinget

 tiny-tim @ 05:14 AM Nov3-12 Corrected mistake in LaTeX.

 Redbelly98 @ 06:42 PM Dec23-08 23 Dec 2008: Edit and save Definition/Summary, Equation, and Extended Explanation sections (no changes) to get rid of LaTex white background.

 blaste @ 11:32 AM Nov25-08 This is de Moivre's Theorem. Euler's identity is achieved by setting x = pi. ~EDIT Disregard that, had a little trouble reading your notation. ~EDIT (tiny-tim) Good idea anyway: de M added.

 ravinayak @ 10:05 PM Nov24-08 i want to know the definition of sin(hx) and cos(hx) ~EDIT(tiny-tim) you mean sinh(x) and cosh(x)? see "exponential"

 Hootenanny @ 06:36 PM Nov24-08 Added "Setting x = 0 we find that this constant must be 1"