Euler Formula: e^(pi*i) = -1

Date: 6/5/96 at 23:18:24
From: Lucas W Tolbert
Subject: Euler Formula: e^(pi*i) = -1

Could you please explain why e^(pi*i) = -1?

Date: 6/6/96 at 12:28:49
From: Doctor Anthony
Subject: Re: Euler Formula: e^(pi*i) = -1

There is the well known identity that e^(ix) = cos(x) + i.sin(x)
Then let x = pi.  cos(pi) = -1,  sin(pi) = 0
so     e^(i.pi) = cos(pi) + i.sin(pi)
                = -1 + 0    
and so e^(i.pi) = -1

From this you can also write the FAMOUS FIVE equation connecting the 
five most important numbers in mathematics, 0, 1, e, pi, i

           e^(i.pi) + 1 = 0

To show the truth of the identity quoted above we let 
z = cos(x) + i.sin(x)
Then dz/dx = -sin(x)+i.cos(x)
           = i{cos(x)+i.sin(x)}
           = i.z

So  dz/z = i.dx   Now integrate
   ln(z) = i.x + const.    When x=0, z=1 so const=0
   ln(z) = i.x

      z = e^(i.x)

So cos(x) + i.sin(x) = e^(i.x) 
   
-Doctor Anthony,  The Math Forum
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