This shape represents the mathematical function y = x raised to the x power, where x is a real number but y is complex.

The model was printed on the Dimension 3-dimensional printer.

For further information about this shape, continue reading below.

Discussion

This discussion will involve taking exponents. If you are rusty on the rules for taking exponents, see the Exponent Rules section below.

This looks like a very tame function. However, zero raised to the zero power is undefined. So x^x 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.

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 = x^x 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 =
3.375^1/2 =
1.837, -1.837

(3/4)^(3/4) =
((3³)/4³))^1/4 =
(27/64)^1/4 =
.421875^1/4 =
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:

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))

(“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 = (.7071^1.414)*exp(i*pi*(-1.414)*(1 + 2n))
y = .6125*exp(i*pi*(-1.414)*(1 + 2n))

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=x^x 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 x^x as x approaches zero from positive values is equal to 1.

Exponent Rules

x^a = x*x*.... (a times) = x to the ath power
Example: 2⁵ = 2*2*2*2*2 = 32 = two to the fifth power

x^(a+b) = (x^a)(x^b)
Example: 2⁽²⁺³⁾ = 2⁵ = 32 = 4*8 = (2²)(2³)

x^(a-b) = (x^a)/(x^b)
Example: 2^(3-2) = (2³)/(2²) = 8/4 = 2 = 2¹ = 2^(3-2)

x^(-b) = 1/(x^b)
Example: 2^(-3) = 1/(2³) = 1/8

x^(a*b) = (x^a)^b = (x^b)^a
Example: 2^(2*3) = 2⁶ = 64 = 4³ = (2²)³ = 8² = (2³)²

if y = x^a then x = y^(1/a)
Example: 8 = 2³ so 2 = 8^(1/3)

x^(1/b) = bth root of x
Example: 81^(1/4) = (3*3*3*3)^(1/4) = 3 = 4th root of 81

x^(a/b) = (x^a)^(1/b)
Example: 8^(2/3) = (8²)^(1/3) = 64^(1/3) = (4*4*4)^(1/3) = 4
Check: 4^(3/2) = (4*4*4)^1/2 = 64^(1/2) = (8*8)^1/2 = 8

Decimal example:
16^(1.25) = 16^{(1 + .25)} = (16¹)(16^(1/4)) = 16*2 = 32
Alternate: 16^(1.25) = 16^(5/4) = (2⁴)^(5/4) = 2^(4*5/4) = 2⁵ = 32
Check: 32^(.8) = 32^(4/5) = (32⁴)^1/5 = (32*32*32*32)^1/5 = (16*2*16*2*16*2*16*2)^1/5 = (16*16*16*16*16)^1/5 = 16