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Science Section => Science General Discussion => Math and Computers => Topic started by: Solitary on June 11, 2013, 10:08:39 PM

Title: Eulers Equation
Post by: Solitary on June 11, 2013, 10:08:39 PM
:oops:  Sorry about that! I have no idea what happen, or what I was posting, but here is something about it:

The Euler formula, sometimes also called the Euler identity (e.g., Trott 2004, p. 174), states


e^(ix)=cosx+isinx,  
(1)
 

where i is the imaginary unit. Note that Euler's polyhedral formula is sometimes also called the Euler formula, as is the Euler curvature formula. The equivalent expression


ix=ln(cosx+isinx)  
(2)
 

had previously been published by Cotes (1714).

The special case of the formula with x=pi gives the beautiful identity


e^(ipi)+1=0,  
(3)
 

an equation connecting the fundamental numbers i, pi, e, 1, and 0 (zero), the fundamental operations +, ×, and exponentiation, the most important relation =, and nothing else. Gauss is reported to have commented that if this formula was not immediately obvious, the reader would never be a first-class mathematician (Derbyshire 2004, p. 202).

The Euler formula can be demonstrated using a series expansion

e^(ix) = sum_(n=0)^(infty)((ix)^n)/(n!)
(4)
 
 = sum_(n=0)^(infty)((-1)^nx^(2n))/((2n)!)+isum_(n=1)^(infty)((-1)^(n-1)x^(2n-1))/((2n-1)!)
(5)
 
 = cosx+isinx.
(6)
 

It can also be demonstrated using a complex integral. Let

z = costheta+isintheta
(7)
 
dz = (-sintheta+icostheta)dtheta
(8)
 
 = i(costheta+isintheta)dtheta
(9)
 
 = izdtheta
(10)
 
int(dz)/z = intidtheta
(11)
 
lnz = itheta,
(12)
 

so

z = e^(itheta)
(13)
 
 = costheta+isintheta.
(14)
 

A mathematical joke asks, "How many mathematicians does it take to change a light bulb?" and answers "-e^(ipi)" which, of course, equals (1)
 :-D  Solitary
Title: Re: Eulers Equation
Post by: Atheon on August 02, 2013, 03:10:29 AM
You mean this?

e^(i*pi) = -1
Title: Re: Eulers Equation
Post by: Colanth on August 02, 2013, 03:29:56 PM
Random data is random, but data sets aren't random data.
Title: Re: Eulers Equation
Post by: Poison Tree on August 02, 2013, 05:56:04 PM
:goodman:
Title: Re: Eulers Equation
Post by: Solitary on August 02, 2013, 06:40:14 PM
Quote from: "Atheon"You mean this?

e^(i*pi) = -1



 :-k  Yes!  #-o   :Hangman:   Solitary
Title: Re: Eulers Equation
Post by: Jason78 on August 03, 2013, 04:32:39 AM
Copied and pasted from Euler Formula (//http://mathworld.wolfram.com/EulerFormula.html)

Nice
Title: Re: Eulers Equation
Post by: josephpalazzo on August 03, 2013, 04:55:14 AM
Quote from: "Solitary"The special case of the formula with x=pi gives the beautiful identity

e^(ipi)+1=0,  
(3)
 

On my blog The Unruh Effect (//http://soi.blogspot.ca/2013/07/the-unruh-effect.html), equation (34).

 :-D