Thanks for your question.
There are many ways of answering this kind of question and many levels
of detail into which one can delve. I will give one answer here, and,
if you like, just let me know if there is any more information or
greater clarification you would like over any particular point and I
would be happy to answer it (use the "Request Clarification" button to
ask for more clarification).
The formula
i theta
e = cos (theta) + i sin(theta)
is indeed one of the most important and beautiful formulas in
mathematics. Indeed, if we substitute the value pi for theta, we get
i pi
e = cos (pi) + i sin(pi)
= -1 + i 0
= -1
(Remember we are measuring theta in radians, so that 2pi radians = 360
degrees.)
I remember when I first saw this formula as a kid how impressed I was.
It was one of the things that made me more curious about higher
mathematics actually.
This formula, "Euler's identity" relating three fundamental constants
in mathematics (e, i, and pi --- four if you count -1) has been called
"the most remarkable formula in the world. "Euler's identity"
http://www.wikipedia.org/wiki/The_most_remarkable_formula_in_the_world
points to "Most remarkable theorem in the world" .
The goal of this short essay is first, to prove the formula; and
second, to discuss its relevance to mechanics.
-------------------
PROVING THE FORMULA
In order to prove the formula, it is necessary first to define "e" and
second to define the notion of raising e to the power of a complex
number . As is virtually always the case for this sort of thing, the
definitions of the main terms have undergone considerable modification
over the centuries. I will give a somewhat more modern treatment here.
However, I will try and motivate the definitions a little bit.
One way to define e is to start by defining the exponential function
x
e
One can define
x 2 3 n
e = 1 + x + x + x + ... + x + ...
--- --- ---
2! 3! n!
Here, n!, or n factorial, denotes 1*2* ... *n, the product of the
first n numbers. 0! is defined to be 1.
More compactly, this is sometimes written:
x j=infinity j
e = Sum x
j=0 ----
j!
Now, in order to prove Euler's formula, we need to know a few facts
about Taylor series.
Every real function (well, every "well-behaved", ) function f(x) can
be written as a an infinite-degree polynomial in f:
j=infinity j
f(x) = Sum a x
j=0 j
Here x is a real number and each of the a's are also real numbers.
The value of each of the a_j's can be computed using calculus, using
the so-called Taylor formula.
(n)
a = f (0)
j ------
n!
where
(n)
f (x) is the value of the n'th derivative of f at 0.
----------Side topic: derivatives--------------
I'm not sure how much about calculus you know, so I will briefly
define a derivative.
Given a nice real function f, we can compute the derivative of f at a
point x. It is written
f'(x)
or
(1)
f (x)
in the notation above.
The derivative is the *rate of change* of f at a particular point.
Formally,
f(x+h)-f(x)
f'(x) = lim -------------
h->0 x-h
For examples, suppose a car is moving in straight line to the right.
Let f(t) be the distance of the car in feet from its starting location
after t seconds. Then
f'(t)
will equal the speed, in feet-per-second of the car at time t.
Also,
f''(t)
will equal the acceleration of the time at time t.
Here are three facts about derivatives:
n
1. If f(x) = x , for some positive integer n, then
n-1
f'(x) = n x
2. If f(x)=u(x) + v(x), then f'(x)=u'(x)+v'(x) .
3. If f(x) = sin(x) then f'(x)=cos(x) .
4. If f(x)=cos(x) then f'(x)= -sin(x) .
Now, using facts 3 and 4, we can derive the Taylor series for sin(x)
and cos(x) .
That's because the derivatives of, say, sin(x) keep repeating, we get
sin(x) has derivative of:
cos(x) which has derivative:
-sin(x) with derivative:
-cos(x) with derivative:
sin(x) and we are back to the beginning.
Anyway, when one carries out the computation for the Taylor series of
sin and cos, one gets this result:
sin(x) =
1 3 1 5 1 7
x - --- x + ---- x - ---- x + ...
3! 5! 7!
That is
sin(x) =
j=infinity j 1 (2j+1)
Sum (-1) * -------- x
j=0 (2j+1)!
Similarly, the Taylor series for cos(x) is:
1 2 1 5 1 7
1- --- x + ---- x - ---- x + ...
2! 5! 7!
with general term of:
j=infinity j 1 (2j)
Sum (-1) * -------- x
j=0 (2j)!
Now, it is easy to prove Euler's formula from the general expression
for e using the Taylor formula for e above.
We use the fact that i*i= -1.
So,
i*t
e
=
2 3 4 5 6
1+(i t)+(i t)+ (i t) + (i t) + (i t) + (i t) + ...
--- --- ---- --- ---
2! 3! 4! 5! 6!
n n n
If we examine a numerator ( i t ) = i t =
n
t if 4 divides n evenly
n
i t if n divided by 4 leaves a remainder of 1
n
- t if n divided by 4 leaves a remainder of 2
n
-i t if n divided by 4 leaves a remainder of 3
it
So, when you plug this back into the equation for e you get:
it
e
2 3 4 5
= t t t t
1 + it - --- - i ---- + ----- + i----- ...
2! 3! 4! 5!
Finally, for the big finish, if you collect all the imaginary terms
together, you find that they equal
i sin(t)
using the Taylor series for sin.
it
Similarly the sum of all the real terms in the expression for e is
cos(t) .
So
it
e = cos(t) + i sin(t) .
This is the proof.
------Importance of exponential function to mechanics-----
z
The importance of the exponential function (that is, e ) in
mechanics, and indeed in all of physics, lies in the following crucial
property of
x
e
Namely, its derivative equals itself.
x
That is, if f(x) = e , then
f'(x) = f(x) .
x
The only other function with this property is multiple of f, like 2*e
.
x
But e also satisfies many other nice properties like
0
e =1
x+y x y
e = e e
It's the only function with these properties whose derivative is equal
to itself.
The reason that this is so important is that the laws of mechanics are
all differential equations. A differential equation is some expression
that relates the values of a function to the values of its
derivatives.
Specifically, Newton's laws are differential equations (it's not
coincidence that Newton was involved so much in both the development
of physics and of calculus).
For example, consider Newton's law:
F = m a ,
Force is mass times acceleration. Well, we saw above that acceleration
is the second derivative of position , so solving differential
equations is fundamental to solving mechanics problems.
The exponential function is crucial to solving differential equations
because it is an identity for them, in the same way that the number 0
is crucial for solving addition problems, or the number 1 for solving
multiplication problems.
Any time there is oscillatory motion in mechanics, in particular, the
exponential function arises.
But, when we solve differential equations in terms of complex
exponentials, we have to find some way to interpret the results in
terms of real numbers. The Euler equation here gives us that formula.
Let's look at a simple explanation for the power of the exponential.
Remember in trigonometry where one learned some complex formula for
sin(x+y) .
Well, now it is trivial to derive, since
sin(x+y) =
i(x+y)
Im( e )
ix iy
= Im (e e )
= Im ( (cos x + i sin x) (cos y + i sin y)
= cos x sin y + sin x cos y .
Similarly, one can get really quick formulas for sin(5 x) for example,
that would be a major hassle to derive without using exponential.
By the way, it is worth noting that in quantum mechanics just about
everything is complex: according to quantum mechanics every particle
has an associated (complex) wave function that pretty much determines
everything about that particle. So in quantum mechanics, exponentials
are even more important.
LINKS
------
Brief discussion of Euler's formula:
http://www.ma.iup.edu/projects/CalcDEMma/complex/complex4.html
Euler formula again:
http://www.wikipedia.org/wiki/Eulers_formula_in_complex_analysis
"Euler's identity" http://www.wikipedia.org/wiki/The_most_remarkable_formula_in_the_world
points to "Most remarkable theorem in the world"
"History of the exponential function", supposedly proved by Cotes in
1714, and in 1740-1748, Euler showed the series expansion.
Plot of complex exponential function:
http://www.kfunigraz.ac.at/imawww/vqm/pages/complex/05_exp.html
10000 digits of e here:
http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/e_10000.html
(Always useful for increasing the page count).
Sin series and cos series:
http://www.ms.uky.edu/~carl/ma330/sin/sin1.html |