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The definition and domain of exponentiation has been changed several times. The original operation x^y was only defined when ywas a positive integer. The domain of the operation ofexponentation has been extended, not so much because the originaldefinition made sense in the extended domain, but because there were(almost) unique ways to extend exponentation which preserved many ofwhat seemed to be the ``important" properties of the original operation.So in part, these definitions are only convention, motivated byreasons 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 onlya reasonable definition for y = 1, 2, 3, ... (It could be argued that itis reasonable when y=0, but that issue is taken up in a different partof the FAQ). This operation has a number of properties, including
- x^1 = x
- For any x, n, m, x^n x^m = x^(n + m).
- If x is positive, then x^n is positive.
Now, we can try to see how far we can extend the domain ofexponentiation so that the above properties (and others) still hold. Thisnaturally leads to defining the operation x^y on the domain x positivereal; y rational, by setting x^(p/q) = the q^(th) root of x^p. Thisoperation agrees with the original definition of exponentiation on theircommon domain, and also satisfies (1), (2) and (3). In fact, it is theunique operation on this domain that does so. This operation also hassome other properties:
- If x>1, then x^y is an increasing function of y.
- 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 exponentiationwhile still preserving properties (1)-(5). This leads naturally to thefollowing 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 allrationals less than or equal to y.
If x<1, x^y is defined to be inf_q { x^q } , where q runs over allrationals bigger than or equal to y.
If x=1, x^y is defined to be 1.
See Alsoe Calculator | eˣ | e Raised to Power of x1.6: Euler's Formula3Blue1Brown - What's so special about Euler's number e?Value of e in Maths (Constant e - Euler's Number)Again, this operation satisfies (1)-(5), and is in fact the only operationon this domain to do so.
The next extension is somewhat more complicated. As can be proved usingthe 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,
- e^x = 1 + x/1! + x^2/2! + x^3/3! + ...
e^x is also denoted exp(x). (This series always converges regardless ofthe value of x).
One can also define an operation ln(x) on the positive reals, which is theinverse of the operation of exponentiation by e. In other words, exp(ln(x)) = x for all positive x. Moreover,
- If x is positive, then x^y = exp(y ln(x)).Because of this, the natural extension of exponentiation to complexexponents, 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 complexwhich satisfies all of (1)-(7). Because of this and other reasons, itis 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 thetrigonometric sine and cosine functions respectively.From this, one sees that, for any real x,
- 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 cosinefrom (8) and (1).
All of the above extensions have been restricted to a positive real forthe base. When the base x is not a positive real, it is not asclear-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 giveinfinitely many candidates for x^y, all equally plausible. And ofcourse x=0 has its own special problems. These problems can all betraced to the fact that the exp function is not injective on the complexplane, so that ln is not well defined outside the real line. There areways around these difficulties (defining branches of the logarithm, forexample), but we shall not go into this here.
The operation of exponentiation has also been extended to other systemslike matrices and operators. The key is to define an exponentialfunction by (6) and work from there. [Some reference on operatorcalculus and/or advanced linear algebra?]
References
Complex Analysis. Ahlfors, Lars V. McGraw-Hill, 1953.
Next: What is 0^0Up: Special Numbers and Functions Previous: How to compute digits
Alex Lopez-Ortiz
Fri Feb 20 21:45:30 EST 1998