Is $x^x$ a polynomial, an exponential or both?

If $c$ is a constant, and $x$ is a variable, we'd say that $f(x) = x^c$ is a polynomial function of order $c$. Conversely, the function $f(x) = c^x$ would be called an exponential function.

Is there a name for a function of the form $f(x) = x^x$? Strictly speaking it's neither an exponential nor a polynomial.

• "Strictly speaking": even loosely speaking, it's neither an exponential nor a polynomial. Here is a related question. Aug 12, 2014 at 21:37

It's neither. A poynomial is a function that is of the form $\sum_i c_ix^i$ where the $c_i$ are constants. An exponential function is one of the form $Ca^x$ for some constant $a$ and nonzero constant $C$ Note that $x$ is not a constant, and so $x^x$ is of neither form.

$$x^x=e^{\ln x^x}=e^{x \ln x}$$

Therefore, it is a composition of an exponential and the product of $x \cdot \ln x$

• Note: the domain of $x^x$ is $x\ne 0$, the domain of $e^{x\ln x}$ is $x>0$. Aug 12, 2014 at 21:06
• The domain of $x^x$ is also $x>0$.
– user159870
Aug 12, 2014 at 21:08
• @Rainier van Es: If you want $x^x$ to be real-valued then its domain is also $x > 0$. Otherwise how do you define $(-\frac{1}{2})^{-\frac{1}{2}}$, for instance? The expressions $x^x$ and $e^{x\ln x}$ are equivalent. Aug 12, 2014 at 21:08
• Thanks for the explanation! Aug 12, 2014 at 21:09
• Interesting, I hadn't thought of it that way. Aug 13, 2014 at 0:39

No, this is not a polynomial as a polynomial necessarily has non-negative integer exponents. I've heard it referred to as a "hyperpower" function and you can read about it, and similar topics, on this page.

Also, you could think of this function as $e^{x\log x}$ which is a composition of polynomial, exponential and logarithmic functions.

• A polynomial certainly need not have non-negative integer coefficients. Perhaps you mean that a polynomial in $x$ has terms whose powers of $x$ are non-negative integers? Aug 12, 2014 at 21:12
• I meant exponents. Thanks for the catch. Aug 12, 2014 at 21:13