Here 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.
When I hear music, I fear no danger. I am invulnerable. I see no foe. I am related to the earliest times, and to the latest.
- Henry David Thoreau -
Condensing fact from the vapor of nuance since 2003
15.4.07
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