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From: alopez-o@neumann.uwaterloo.ca (Alex Lopez-Ortiz)
Newsgroups: sci.math,news.answers,sci.answers
Subject: sci.math FAQ: Euler's Formula
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Date: 17 Feb 2000 22:51:58 GMT
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Archive-name: sci-math-faq/euler
Last-modified: February 20, 1998
Version: 7.5
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Euler's formula: e^(i pi) = -1
The definition and domain of exponentiation has been changed several
times. The original operation x^y was only defined when y was a
positive integer. The domain of the operation of exponentation has
been extended, not so much because the original definition made sense
in the extended domain, but because there were (almost) unique ways to
extend exponentation which preserved many of what seemed to be the
``important" properties of the original operation. So in part, these
definitions are only convention, motivated by reasons of aesthetics
and utility.
The original definition of exponentiation is, of course, that x^y = 1
* x *x * ... * x, where 1 is multiplied by x, y times. This is only a
reasonable definition for y = 1, 2, 3, ... (It could be argued that it
is reasonable when y=0, but that issue is taken up in a different part
of the FAQ). This operation has a number of properties, including
1. x^1 = x
2. For any x, n, m, x^n x^m = x^(n + m).
3. If x is positive, then x^n is positive.
Now, we can try to see how far we can extend the domain of
exponentiation so that the above properties (and others) still
hold. This naturally leads to defining the operation x^y on the
domain x positive real; y rational, by setting x^(p/q) = the
q^(th) root of x^p. This operation agrees with the original
definition of exponentiation on their common domain, and also
satisfies (1), (2) and (3). In fact, it is the unique operation on
this domain that does so. This operation also has some other
properties:
4. If x>1, then x^y is an increasing function of y.
5. If 01, x^y is defined to be sup_q { x^q } , where q runs over a
ll rationals less than or equal to y.
If x<1, x^y is defined to be inf_q { x^q } , where q runs over a
ll rationals bigger than or equal to y.
If x=1, x^y is defined to be 1.
Again, this operation satisfies (1)-(5), and is in fact the only
operation on this domain to do so.
The next extension is somewhat more complicated. As can be proved
using the methods of calculus or combinatorics, if we define e to
be the number
e = 1 + 1/1! + 1/2! + 1/3! + ... = 2.71828...
it turns out that for every real number x,
6. e^x = 1 + x/1! + x^2/2! + x^3/3! + ...
e^x is also denoted exp(x). (This series always converges
regardless of the value of x).
One can also define an operation ln(x) on the positive reals,
which is the inverse of the operation of exponentiation by e. In
other words, exp(ln(x)) = x for all positive x. Moreover,
7. If x is positive, then x^y = exp(y ln(x)). Because of this, the
natural extension of exponentiation to complex exponents, seems to
be to define
exp(z) = 1 + z/1! + z^2/2! + z^3/3! + ...
for all complex z (not just the reals, as before), and to define
x^z = exp(z ln(x))
when x is a positive real and z is complex.
This is the only operation x^y on the domain x positive real, y
complex which satisfies all of (1)-(7). Because of this and other
reasons, it is accepted as the modern definition of
exponentiation.
From the identities
sin x = x - x^3/3! + x^5/5! - x^7/7! + ...
cos x = 1 - x^2/2! + x^4/4! - x^6/6! + ...
which are the Taylor series expansion of the trigonometric sine
and cosine functions respectively. From this, one sees that, for
any real x,
8. exp(ix) = cos x + i sin x.
Thus, we get Euler's famous formula
e^(pi i) = -1
and
e^(2 pi i) = e^0 = 1.
One can also obtain the classical addition formulae for sine and
cosine from (8) and (1).
All of the above extensions have been restricted to a positive real
for the base. When the base x is not a positive real, it is not as
clear-cut how to extend the definition of exponentiation. For example,
(-1)^(1/2) could well be i or -i, (-1)^(1/3) could be -1, 1/2 +
sqrt(3)i/2, or 1/2 - sqrt(3)i/2, and so on. Some values of x and y
give infinitely many candidates for x^y, all equally plausible. And of
course x=0 has its own special problems. These problems can all be
traced to the fact that the exp function is not injective on the
complex plane, so that ln is not well defined outside the real line.
There are ways around these difficulties (defining branches of the
logarithm, for example), but we shall not go into this here.
The operation of exponentiation has also been extended to other
systems like matrices and operators. The key is to define an
exponential function by (6) and work from there. [Some reference on
operator calculus and/or advanced linear algebra?]
References
Complex Analysis. Ahlfors, Lars V. McGraw-Hill, 1953.
--
Alex Lopez-Ortiz alopez-o@unb.ca
http://www.cs.unb.ca/~alopez-o Assistant Professor
Faculty of Computer Science University of New Brunswick