Does this distribution have a name? $f(x)\propto\exp(-|x-\mu|^p/\beta)$ It occurred to me today that the distribution 
$$
f(x)\propto\exp\left(-\frac{|x-\mu|^p}{\beta}\right)
$$
could be viewed as a compromise between the Gaussian and Laplace distributions, for $x\in\mathbb{R}, p\in[1,2]$ and $\beta>0.$ Does such a distribution have a name? And does it have an expression for its normalization constant? The calculus stumps me, because I don't know how to even begin solving for $C$ in the integral
$$
1=C\cdot \int_{-\infty}^\infty \exp\left(-\frac{|x-\mu|^p}{\beta}\right) dx 
$$
 A: Short Answer 
The pdf you describe is most appropriately known as a Subbotin  distribution ... see the paper in 1923 by Subbotin which has exactly the same functional form, with say $Y = X-\mu$.


*

*Subbotin, M. T. (1923), On the law of frequency of error, Matematicheskii Sbornik, 31, 296-301.


who enters the pdf at his equation 5, of form:
$$f(y) = K \exp\left[-\left(\frac{|y|}{\sigma}\right)^p\right]$$
with constant of integration:  $K = \large\frac{p}{2 \sigma \Gamma \left(\frac{1}{p}\right)}$, as per Xian's derivation where $\beta = \sigma^p$ 
Longer answer
Wikipedia is unfortunately not always 'up to date', or accurate, or sometimes just 80 years behind the times.  After Subbotin (1923), the distribution has been widely used in the literature, including:


*

*Diananda, P.H. (1949), Note on some properties of maximum likelihood estimates, Proceedings of the Cambridge Philosophical Society, 45, 536-544.

*Turner, M.E. (1960), On heuristic estimation methods, Biometrics, 16(2), 299-301.

*Zeckhauser, R. and Thompson, M. (1970), Linear regression with non-normal error terms, The Review of Economics and Statistics, 52, 280-286.

*McDonald, J.B. and Newey, W.K. (1988), Partially adaptive estimation of regression models via the generalized t distribution, Econometric Theory, 4, 428-457.

*Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995), Continuous Univariate Distributions, volume 2, 2nd edition, Wiley: New York (1995, p.422)

*Mineo, A.M. and Ruggieri, M. (2005), A software tool for the Exponential Power distribution: the normalp package, Journal of Statistical Software, 12(4), 1-21.
... all before the paper referenced on Wiki. Aside from being 80 years out of date, the name used on Wiki 'a Generalised Normal' also seems inappropriate because there are an infinity of distributions that are generalisations of the Normal, and the name is, in any event, ambiguous to the literature. It also fails to acknowledge the original author.
A: For obvious reasons, you can get rid of μ and β so all that remains is $$\int_0^\infty \exp\{−x^p\}\text{d}x\stackrel{y=x^p}{=}\int_0^\infty \exp\{−y\}\left|\dfrac{\text{d}x}{\text{d}y}\right|\text{d}y\stackrel{x=y^{1/p}}{=}\int_0^\infty \exp\{−y\}\frac{1}{p}y^{\frac{1}{p}-1}\text{d}y=\Gamma(1/p)\frac{1}{p}
$$
Hence
$$\int_{-\infty}^\infty \exp\{−\beta^{-1}|x-\mu|^p\}\text{d}x=\dfrac{2\Gamma(1/p)}{p}\beta^{1/p}$$
A: According to Wikipedia, this is known as  Generalized normal distribution (version 1 in the article), and the restriction $p\in [1,2]$ is not required but any positive value is fine. 
The reference given in Wikipedia is  Saralees Nadarajah (2005) A generalized normal distribution, Journal of
Applied Statistics, 32:7, 685-694, DOI: 
10.1080/02664760500079464. This article mentions that the normalization constant is found by 'simple integration' - I presume following Xi'an's answer.
