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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.

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

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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.

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

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

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    $\begingroup$ Note: the domain of $x^x$ is $x\ne 0$, the domain of $e^{x\ln x}$ is $x>0$. $\endgroup$
    – rae306
    Aug 12, 2014 at 21:06
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    $\begingroup$ The domain of $x^x$ is also $x>0$. $\endgroup$
    – user159870
    Aug 12, 2014 at 21:08
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    $\begingroup$ @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. $\endgroup$
    – Clive Newstead
    Aug 12, 2014 at 21:08
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    $\begingroup$ Thanks for the explanation! $\endgroup$
    – rae306
    Aug 12, 2014 at 21:09
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    $\begingroup$ Interesting, I hadn't thought of it that way. $\endgroup$
    – Anthony E
    Aug 13, 2014 at 0:39
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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.

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  • $\begingroup$ 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? $\endgroup$
    – qaphla
    Aug 12, 2014 at 21:12
  • $\begingroup$ I meant exponents. Thanks for the catch. $\endgroup$
    – MRicci
    Aug 12, 2014 at 21:13

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