Graph of y = x^x

1. May 21, 2006

meee

Heyyhey...just wondering, is the graph of y = x^x significant in anyway?

it looks kinda weird...?

2. May 21, 2006

arildno

Chinese cooks usually put the spaghetti threads in that particular shape on your plate.

Other than that, I don't know if that graph is "significant".

3. May 21, 2006

benorin

it is particularly weird for x<0, being that it takes complex values there...

4. May 21, 2006

arildno

The Chinese have never liked the negatives.

5. May 22, 2006

Dragonfall

I can't get mathematica to plot this function for negative values. Anyone know how I can do it?

6. May 22, 2006

arunbg

7. May 22, 2006

Dragonfall

I fail to see how I would be unable to plot it.

8. May 22, 2006

arunbg

Complex meaning they are imaginary.
Try x=-1/2

9. May 22, 2006

heartless

Yep, just do Plot[{y=x^2},{x,-10,10}] and you get all the values from -10 to 10 of this super significant function x^x.

10. May 22, 2006

Dragonfall

Mathematica can plot complex functions, and this function in particular because it's R->C.

How does plotting x^2 give me all the values of x^x?

Last edited: May 22, 2006
11. May 23, 2006

meee

thnx cool guys... whats the derivative of y=x^x ?

12. May 23, 2006

LeonhardEuler

$$y=x^x=e^{\ln{x^x}}=e^{x\ln{x}}$$
$$\frac{dy}{dx}=(1+\ln{x})e^{x\ln{x}}=(1+\ln{x})x^x$$

13. May 23, 2006

Curious3141

It is much more interesting and informative to plot the hyperpower function $$f(x) = x^{x^{x^{x^{...}}}}$$

Find the upper bound of x for which that function is defined and see if you can spot the relationship of that bound to a famous constant.

Last edited: May 23, 2006
14. May 24, 2006

benorin

$$\frac{dy}{dx}=x^x(1+\ln{x})$$ is not real when x is a real negative number, yet if x is negative and of the form $$x=\frac{p}{2q+1}$$, where p,q are positive or negative integers, then y is real. Curious, no? It has to do with the complex branch-cut structure of $$y=x^x=e^{x\ln{x}+2k\pi ix},k=0,\pm 1, \pm 2,\ldots$$.

Source: "A Course of Modern Analysis" by Whittaker & Watson, pg. 107.

15. May 25, 2006

Curious3141

I'm not sure about this, certainly for x in that domain and of that form, a real value of y exists if y is defined to be multivalued.

But if the principal value of ln(x) is used, which is $$Ln(x) + \pi i$$, then the value of y returned won't necessarily be real, right?

Sorry, I don't know that much about complex analysis, just the basics. I do know the principal branch for ln x, but not the one for x^x. I would've assumed it would be based on the branch cut of the log function, giving $$x^x = e^{x Ln(x)} = e^{x ln(|x|) + i\pi x} = \frac{\cos{(\pi |x|)} - i\sin{(\pi |x|)}}{|x|^{|x|}}$$ (for negative real x) which would not necessarily return real values even for x of the form $$\frac{n}{2k+1}$$

Last edited: May 25, 2006
16. Jun 6, 2006

I've always liked that the only critical point of $$f(x) = x^x$$ is
$$x = \frac{1}{e}$$