: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
You mean this?
e^(i*pi) = -1
Random data is random, but data sets aren't random data.
:goodman:
Quote from: "Atheon"You mean this?
e^(i*pi) = -1
:-k Yes! #-o :Hangman: Solitary
Copied and pasted from Euler Formula (//http://mathworld.wolfram.com/EulerFormula.html)
Nice
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