Drexel dragonThe Math ForumDonate to the Math Forum

Ask Dr. Math - Questions and Answers from our Archives
_____________________________________________
Associated Topics || Dr. Math Home || Search Dr. Math
_____________________________________________

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/   
    
Associated Topics:
High School Imaginary/Complex Numbers
High School Transcendental Numbers
High School Trigonometry

Search the Dr. Math Library:


Find items containing (put spaces between keywords):
 
Click only once for faster results:

[ Choose "whole words" when searching for a word like age.]

all keywords, in any order at least one, that exact phrase
parts of words whole words

Submit your own question to Dr. Math

[Privacy Policy] [Terms of Use]

_____________________________________
Math Forum Home || Math Library || Quick Reference || Math Forum Search
_____________________________________

Ask Dr. MathTM
© 1994-2013 The Math Forum
http://mathforum.org/dr.math/