What is the name of this function?

[Rectifier]

This is not a problem from a book. It is something I am wondering.

$f(x)=x^a$ is an exponential function
$f(x)=a^x$ is a power function
$f(x)=x$ is a identity function

What is $f(x)=x^x$ called then?

 

[Fredrik]

I don't think there's a special name for it. Since $x^x=e^{\log x^x}= e^{x\log x}$, it's the composition of an exponential function and the product of the identity function and a logarithm.

Oh, and what you called a power function is an exponential function and vice versa. The term "exponential function" is used when the independent variable (in this case x) is in the exponent.

 

[andrewkirk]

As Fredrik says, there is no name for that specific one. However, it is a particular case of a more general function called 'superexponentiation' or 'tetration', which is the next step in the chain of generalisation that goes from incrementation (by 1) to addition to multiplication to exponentiation.

Tetration, denoted by ${}^nx$ or $x\uparrow\uparrow n$ is a number raised to itself as power n times. The one you have written is denoted in Tetration notation by ${}^2x$ or $x\uparrow\uparrow 2$ (the latter is Donald Knuth's notation, which was designed to be amenable for easy further generalisation). Note that the calculation needs to be done in the order from top right towards bottom left, otherwise you just end up with $x^{(n-1)x}$.

You can read more about Tetration here.

As you would expect, the process of generalisation and extension can be continued indefinitely many times to get even bigger operations called hyperoperations. This leads to the fascinating (but not terribly useful) study of Very Large Numbers like Graham's Number and beyond.

 

[Rectifier]

Thank you for your help!