Euler Equation

Date: 01/21/97 at 23:45:40
From: Alan Davis
Subject: formula for e

I am sure that  e^(pi*i) = -1 has no meaning, but I'd like to follow 
the steps. 

Date: 01/22/97 at 18:54:58
From: Doctor Mitteldorf
Subject: Re: formula for e

Dear Alan,

e^i*pi = -1 is a special case of the Euler equation

e^i*x = cos(x) + i*sin(x)

This equation, it turns out, is enormously interesting and useful.  
It's not arbitrary, in the sense that once you decide to define 
sqrt(-1) = i, the Euler equation must be true.

Here are some of the things you can do to attach some meaning in your 
own mind to the Euler equation:

1. Write down the infinite series 
     e^x = 1 + x + x^2/2! + x^3/3! + x^4/4! + ...
   Write down the series for sin(x) and cos(x)
     sin(x) = x - x^3/3! + x^5/5! + ...
     cos(x) = 1 - x^2/2! + x^4/4! + ...
   Use them to verify the Euler equation.

2. Use the trig formulas for cos(2x) and sin(2x) to find 
   cos(2x) + i*sin(2x)

Now, if this function (for e) is really an exponential, it should be 
growing exponentially.  In other words, when you take the function 
with 2x as argument, you should just get the square of what you get
when x is the argument.  When you take the function of 3x, you should
get the cube of the function of x.  Does the function cos(x)+i*sin(x)
have this property?  Test it and see.

3. If you know a little calculus, you can try differentiating the 
Euler equation to see that you get a consistent answer.

There are branches of physics where people are constantly using the
Euler equation to go back and forth between the trigonometric and the 
exponential version of a formula, because first one version then the
other turns out to be easier to work with.  To them, the Euler 
equation has become second nature.

-Doctor Mitteldorf,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/