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

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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?
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```
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
Check out our web site!  http://mathforum.org/dr.math/
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Associated Topics:
College Analysis
High School Analysis
High School Transcendental Numbers

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