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Eulers Equation

Started by Solitary, June 11, 2013, 10:08:39 PM

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Solitary

: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
There is nothing more frightful than ignorance in action.

Atheon

You mean this?

e^(i*pi) = -1
"Religion is regarded by the common people as true, by the wise as false, and by the rulers as useful." - Seneca

Colanth

Random data is random, but data sets aren't random data.
Afflicting the comfortable for 70 years.
Science builds skyscrapers, faith flies planes into them.

Poison Tree

"Observe that noses were made to wear spectacles; and so we have spectacles. Legs were visibly instituted to be breeched, and we have breeches" Voltaire�s Candide

Solitary

Quote from: "Atheon"You mean this?

e^(i*pi) = -1



 :-k  Yes!  #-o   :Hangman:   Solitary
There is nothing more frightful than ignorance in action.

Jason78

Winner of WitchSabrinas Best Advice Award 2012


We can easily forgive a child who is afraid of the dark; the real
tragedy of life is when men are afraid of the light. -Plato

josephpalazzo

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, equation (34).

 :-D