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sci.math FAQ: Euler's Formula


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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 0<x<1, then x^y is a decreasing function of y.
       Again, we can again see how far we can extend the domain of
       exponentiation while still preserving properties (1)-(5). This
       leads naturally to the following definition of x^y on the domain x
       positive real; y real:
       If x>1, 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

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