This discussion will involve taking exponents. If you are rusty on the rules for taking exponents, read here: Exponent Rules.
| x |
y |
| 1 |
1 |
| 2 |
4 |
| 3 |
27 |
| 4 |
256 |
| 5 |
3125 |
This looks like a very tame function. However, zero raised to the zero power is undefined. So xx for x=0 will remain uncalculated. See the conclusions below.
Recalling that a negative exponent represents reciprocation, we can extend the table to negative values of x.
| x |
y |
| -1 |
-1 |
| -2 |
1/4 |
| -3 |
-1/27 |
| -4 |
1/256 |
| -5 |
-1/3125 |
Notice that the function appears to oscillate for negative values of x. If this is the case, one might expect that there would be a value of x between -1 and -2 where y is zero.
Let’s do the math for x = -3/2.
(-3/2)(-3/2) =
(-2/3)(3/2) =
((-2/3)3)1/2 =
((-23)/(33))1/2 =
(-8/27)1/2 =
(-0.2963)1/2
Recalling that an exponent of 1/2 indicates a square root, then there are two solutions. Also, square root of a negative number causes a rotation of ninety degrees in the complex plane, indicated by the letter ‘i’.
(-8/27)1/2 =
(-0.2963)1/2 =
0.544331i, -0.544331i
So there’s the answer: The function y = xx becomes complex for certain values. Here are a couple of examples for positive x values:
(3/2)(3/2) =
((33)/(23))1/2 =
(27/8)1/2 =
3.3751/2 =
1.837, -1.837
(3/4)(3/4) =
((33)/43))1/4 =
(27/64)1/4 =
.4218751/4 =
0.8059, 0.8059i, -0.8059, -0.8059i
The previous example has four roots. This can be shown to be a general rule that the number of complex roots equals value of the denominator of the exponent.
An alternative form is to use the polar form, y = R*exp(i*theta), where R is the magnitude, “exp” is the exponential function, and theta is the angle in the complex plane. Because x is real, x = X*exp(i*pi*(0+2n)) when positive, x = X*exp(i*pi*(1+2n)) when negative, where X = magnitude of x, n= 0,1,...
Then R = X^x and theta = pi*X*(0+2n) or pi*(-X)*(1+2n).
Using that form, the four roots of the previous example become:
| R |
n |
theta |
| 0.8059 |
0 |
0 |
| 0.8059 |
1 |
1/2 pi |
| 0.8059 |
2 |
pi |
| 0.8059 |
3 |
3/2 pi |
As another example, consider x = -1/3.
y = (1/3 exp (i*pi*(1 + 2n)))(-1/3) =
3(1/3) * exp(i*pi*(1+2n)*(-1/3))=
1.44225 * exp(i*pi*(1+2n)*(-1/3))
| R |
n |
theta |
principal theta |
| 1.44225 |
0 |
-1/3*pi |
5/3*pi |
| 1.44225 |
1 |
-1*pi |
1*pi |
| 1.44225 |
2 |
-5/3*pi |
1/3*pi |
(“principal theta” lists theta values between 0 and 2*pi.)
The only remaining thing to examine is the case of an irrational value of x. Let x = -sqrt(2). Find all the values for y in polar form.
y = (1.414*exp(i*pi*(1 + 2n)))(-1.414)
y = (.70711.414)*exp(i*pi*(-1.414)*(1 + 2n))
y = .6125*exp(i*pi*(-1.414)*(1 + 2n))
| R |
n |
theta |
principal theta |
| 0.6125 |
0 |
-1.414*pi |
0.586*pi |
| 0.6125 |
1 |
-4.242*pi |
1.758*pi |
| 0.6125 |
2 |
-7.071*pi |
0.930*pi |
| 0.6125 |
3 |
-9.898*pi |
0.102*pi |
| 0.6125 |
4 |
-12.73*pi |
1.274*pi |
| 0.6125 |
5 |
-15.55*pi |
0.446*pi |
| ... |
... |
... |
... |
This list goes on forever and the principal theta values never repeat. So an irrational x results in a ring of an infinite number of points in the complex-y-plane with radius equal to |x|x.
Because the irrational numbers outnumber the rationals, the graph of y=xx is a 3-dimensional surface with its axis of symmetry along the x axis and with its radius equal to |x|x. The surface is everywhere discontinuous except for the positive real part of y when x is positive.
Although 00 is undefined, we can write: Limit of xx as x approaches zero is equal to 1.
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