X^x, does this function have a name ?

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In summary, the conversation discusses the differentiation of x^x, also known as the "power tower of order 2" function, and its lack of a commonly used name. It is also mentioned that there is no indefinite integral for x^x that can be expressed in terms of "common" special functions. The conversation also includes examples of integrals that Mathematica cannot compute, as well as instances where Mathematica has given incorrect answers. The speaker also brings up the importance of checking the validity of results obtained from calculators or computers.
  • #1
uart
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Hi, the other day I was explaining to somebody the difference between a power function like x^a and an exponential function like a^x (it was in relation to differentiation). So they asked me "what about x^x" and I was stumped to give it a name.

BTW, I was able to differentiate x^x ok, I got ( 1+ln(x) ) x^x which I'm pretty sure is correct, I just don't know if x^x has a commonly used name.
 
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  • #2
Indeed your differentiation is correct. It doesn't have any sort of name that I know of (I've never seen it used for anything, either). A more useful representation is just

[tex]e^{x\ln x},[/tex]

of course.
 
  • #3
It doesn't have a name.The exponential form saves you from learning yet another concept:logarthmic differentiation.

Daniel.
 
  • #4
Data said:
A more useful representation is just [tex]e^{x\ln x},[/tex]

Doh of course. I don't know why I didn't think of that, particularly as I was explaining to this guy how to express a^x as [tex]e^{x\ln a}[/tex] (in order to differentiate it) at that very time. :blushing:
 
  • #5
unfortunately there is no indefinite integral for x^x
 
  • #6
Anectodically : the function [tex] x^x [/tex] is called "power tower of order 2"
 
  • #7
Power tower is defined for an infinity of exponents...So it's a bit improper.

I'm sure that the function [itex] x^{x} [/itex] is Riemann integrable on a subdomain from R...

Daniel.
 
  • #8
Yes, it is, of course (it's continuous on closed subintervals of [itex]\mathbb{R}^+[/itex], and as demonstrated above, differentiable too!). There's no closed-form antiderivative though.
 
  • #9
[tex] \int x^{x} \ dx [/tex] cannot be expressed in terms of "common" special functions...It it were,i'm sure Mathematica would have done it...

Daniel.
 
  • #10
I loved it when I first worked out the minimum point of xx, really brought calc alive for me. I didn't have it set as an example I just realized as soon as I had learned all the needed things that I could do it.
 
  • #11
dextercioby said:
[tex] \int x^{x} \ dx [/tex] cannot be expressed in terms of "common" special functions...It it were,i'm sure Mathematica would have done it...

Proof by Mathematica?
j/k
 
  • #12
Yeah,go the integrator's webpage and type that function.The calculations are made with the latest version of the software.

Daniel.
 
  • #13
dextercioby said:
Yeah,go the integrator's webpage and type that function.The calculations are made with the latest version of the software.

Daniel.
Actually not true, I've now reported a series of integrations to Wolfram on things their online integrator should be able to integrate and can't. But things fairly obscure and I don’t think any people other than very bored ones like me will find it.
 
  • #14
Give an example.You can't contradict me with words,not in math,at least.

Daniel.
 
  • #15
lambert function and product i know have something to do with x^x=y
 
  • #16
as to x^x, I usually called it george, but sometimes ozymandias.

I have occasionally wondered what the name of the following number is as well:

0.01001000100001000001...

i also have found simple partial fractions that mathematica would not do, but which i did myself with pencil and paper.

More importantly to me, when I was teaching graduate algebra and galois theory i used mathematica to do some of the discriminant calculations for solutions formulas of cubics and quadratics, and found that sometimes mathematica just gave the wrong answer.

It did not say it couldn't do it, or that I had enetered it wrong (usually the case) but simply gave a false answer.

The errors I made in entering data were subtle, like omitting a space somewhere, but the upshot was that a naive student, who believed everything he got back, would have been misled by these incorrect responses, just as when using any calculator. So you always have to know how to check the validity of what you get from a calculator or computer.
 
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  • #17
Reminds of my Maple.I had to reverse a fraction in an ODE so he could be able to solve it...

A software is made by a human.Computers beat human mind for speed/efficiency.

Daniel.
 
  • #18
dextercioby said:
Give an example.You can't contradict me with words,not in math,at least.

Daniel.
It can compute the two following:

[tex]\int \frac{1}{e^x + 2 e^{2 x} + 3 e^{3 x}} dx[/tex]

[tex]\int \frac{x^2}{e^x + 2 e^{2 x} + 3 e^{3 x}} dx[/tex]

But not:

[tex]\int \frac{1 + x^2}{e^x + 2 e^{2 x} + 3 e^{3 x}} dx[/tex]
 
  • #19
:smile: Yeah,it's a big black bug there...

Mathematica latest version won't do this

[tex] \int \frac{x+1}{e^{x}+2e^{2x}+3e^{3x}} \ dx [/tex]

A 10 y.o. Maple returns proudly

[tex] \int \frac{x+1}{e^x+2e^{2x}+3e^{3x}}dx=\allowbreak -\frac {1}{e^x}\ln \left( e^x\right) -\frac {2}{e^x}-\ln ^2\left( e^x\right)+\allowbreak i\ln \left( -\frac {1}{3}-\frac {1}{3}i\sqrt{2}\right) \arctan \frac {1}{2}\sqrt{2} [/tex]

[tex]+\frac {1}{8}i\sqrt{2}\ln \left( -\frac {1}{3}+\frac {1}{3}i\sqrt{2}\right) \ln \left( 1+2e^x+3e^{2x}\right) -\allowbreak \frac {1}{4}\sqrt{2}\ln \left( -\frac {1}{3}+\frac {1}{3}i\sqrt{2}\right) \arctan \left( \frac {3}{2}e^x\sqrt{2}+\frac {1}2\sqrt{2}\right) [/tex]

[tex]+\frac {1}{4}\sqrt{2}\ln \left( -\frac {1}{3}+\frac {1}{3}i\sqrt{2}\right) \arctan \frac {1}{2}\sqrt{2}+\allowbreak \frac {1}{2}\ln \left( -\frac {1}{3}+\frac {1}{3}i\sqrt{2}\right) \ln \left( 1+2e^x+3e^{2x}\right) [/tex]

[tex]-\allowbreak i\ln \left( -\frac {1}{3}+\frac {1}{3}i\sqrt{2}\right) \arctan \frac {1}{2}\sqrt{2}-\frac {1}{8}i\sqrt{2}\ln \left( -\frac {1}{3}-\frac {1}{3}i\sqrt{2}\right) \ln \left( 1+2e^x+3e^{2x}\right) [/tex]

[tex]-\allowbreak \frac {1}{4}\sqrt{2}\ln \left( -\frac {1}{3}-\frac {1}{3}i\sqrt{2}\right) \arctan \left( \frac {3}{2}e^x\sqrt{2}+\frac {1}{2}\sqrt{2}\right) +\frac {1}{4}\sqrt{2}\ln \left( -\frac {1}{3}-\frac {1}{3}i\sqrt{2}\right) \arctan \frac {1}{2}\sqrt{2}[/tex]

[tex]+\allowbreak \frac {1}{2}\ln \left( -\frac {1}{3}-\frac {1}{3}i\sqrt{2}\right) \ln \left( 1+2e^x+3e^{2x}\right) -i\ln \left( -\frac {1}{3}-\frac {1}{3}i\sqrt{2}\right) \arctan \left( \frac {3}{2}e^x\sqrt{2}+\frac {1}{2}\sqrt{2}\right) [/tex]

[tex]-\frac {1}{4}i\sqrt{2}\ \mbox{dilog}\left( 3\frac{e^x}{-1+i\sqrt{2}}\right) - \ \mbox{dilog}\left( 3\frac{e^x}{-1+i\sqrt{2}}\right) +\allowbreak \frac {1}{4}i\sqrt{2} \ \mbox{dilog}\left( -3\frac{e^x}{1+i\sqrt{2}}\right) - \ \mbox{dilog}\left( -3\frac{e^x}{1+i\sqrt{2}}\right)[/tex]

[tex] -2\ln \left( e^x\right) +\allowbreak \ln \left( 1+2e^x+3e^{2x}\right) -\frac {1}{2}\sqrt{2}\arctan \frac {1}{4}\left( 2+6e^x\right) \sqrt{2}
+i\ln \left( -\frac {1}{3}+\frac {1}{3}i\sqrt{2}\right) \arctan \left( \frac {3}{2}e^x\sqrt{2}+\frac {1}{2}\sqrt{2}\right)+ C' [/tex]


:approve:

Daniel.
 
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  • #20
dextercioby said:
...
A 10 y.o. Maple returns proudly
...
[the spawn of Satan himself]
...
:approve:

Daniel.

Quick, someone take the derivative and find out whether it's right. :-p
 
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  • #21
hypermorphism said:
Quick, someone take the derivative and find out whether its right. :-p

Learn how to spell words and then be a wise-ass...

Daniel.
 
  • #22
dextercioby said:
Learn how to spell words and then be a wise-ass...

Daniel.

w-o-r-d-s ? :rolleyes:

Seriously, that would be tedious to differentiate by hand (or even to enter into Mathematica).
 
  • #23
[tex] \frac {d}{dx}\ \mbox{dilog} x =\allowbreak \frac{\ln x}{1-x}[/tex]

is all u need to know.There are 22 terms,outta which a few are constant...

Daniel.
 
  • #24
dextercioby said:
:smile: Yeah,it's a big black bug there...

Mathematica latest version won't do this

[tex] \int \frac{x+1}{e^{x}+2e^{2x}+3e^{3x}} \ dx [/tex]

A 10 y.o. Maple returns proudly
Oi!

Don't bad mouth the actual Mathematica, it can integrate that fine, even if the web version can't. Same with the other problem, I was just pointing out that the web version doesn't use the latest software because it can't do things it should be able to, I have found other things as well it can't do.
 
  • #25
tongos said:
lambert function and product i know have something to do with x^x=y

I sometimes think of x^x as being kind of similar to factorial or Gamma. What I mean is it seems to be usful if you're trying to think of a function that is bigger than factorial but not too much bigger.
 
  • #26
[tex]x^x=x\uparrow\uparrow2[/tex] in Knuth's up-arrow terminology; it's [tex]x\rightarrow2\rightarrow2[/tex], as I recall, in Conway's chained arrow notation.
 

Related to X^x, does this function have a name ?

1. What is the significance of using "X^x" in a function?

The notation "X^x" in a function indicates an exponential relationship, where the variable X is raised to the power of itself. This type of function grows at an increasingly rapid rate as the input value increases.

2. Can "X^x" be used to model real-life situations?

Yes, "X^x" can be used to model real-life situations such as population growth, compound interest, and radioactive decay. In these situations, the variable X represents time and the output of the function represents the quantity or amount of the phenomenon being observed.

3. Does this function have a specific name?

Yes, the function "X^x" is commonly known as the exponential function. It is represented by the mathematical notation f(x) = X^x, where f(x) is the output value and X is the input value.

4. What is the domain and range of "X^x"?

The domain of "X^x" is all real numbers, as the variable X can take on any value. The range of "X^x" depends on the base value of X. If X is greater than 1, the range will be all positive real numbers. If X is between 0 and 1, the range will be all positive real numbers between 0 and 1. And if X is negative, the range will be all negative real numbers.

5. Are there any other variations of "X^x"?

Yes, there are other variations of "X^x" such as "a^X", where a is a constant value. This type of function is known as an exponential growth or decay function. There is also the function "X^a", where a is a constant value, known as a power function. These variations have different characteristics and behaviors, but they all involve the exponential relationship between the input and output values.

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