I've come across the following closed-form formulas for the modified Bessel function of the second kind, $K_{\nu}(x)$.
$$K_{1/2}(x) = \sqrt{\frac{\pi}{2x}} e^{-x} \;,$$ and $$K_{3/2}(x) = \sqrt{\frac{\pi}{2x}} e^{-x} \left( 1 + \frac{1}{x} \right) \;.$$
However, I can't seem to find a closed-form formula anywhere for $K_1(x)$. Does anyone happen to know of such a formula, or perhaps a reference for one? Any help would be much appreciated.
$K_\nu(x)$ is a special function. Any special function was defined and standardised because it cannot be expressed with a finite number of elementary functions. One might say that a special function is it's own closed form.
Sometimes a special functions reduces to function(s) of lower level or to elementary functions for particular values of a parameter. But not for any value of the parameter.
This is the case of the function $K_\nu(x)$ when $\nu=n+\frac{1}{2}$ : $$K_{1/2}(x)=\sqrt{\frac{\pi}{2x}}e^{-x}$$ $$K_{3/2}(x)=\sqrt{\frac{\pi}{2x}}e^{-x}\left(1+\frac{1}{x}\right)$$ $$K_{5/2}(x)=\sqrt{\frac{\pi}{2x}}e^{-x}\left(1+\frac{3}{x}+\frac{3}{x^2}\right)$$ $$K_{7/2}(x)=\sqrt{\frac{\pi}{2x}}e^{-x}\left(1+\frac{6}{x}+\frac{15}{x^2}+\frac{15}{x^3}\right)$$ etc.
But this is not the case for integer $\nu=n$. For example $K_1(x)$ is it's own closed form.
This kind of behaviour is common for a lot of special functions. For example with the Bessel function of first kind $J_\nu(x)$ : $$J_{1/2}(x)=\sqrt{\frac{2}{\pi}}\sin(x)$$ $$J_{3/2}(x)=\sqrt{\frac{2}{\pi x}}e^{-x}\left(\frac{\sin(x)}{x}-\cos(x)\right)$$
Other examples with the Incomplete Gamma function $\Gamma(\nu,x)$ and integer $\nu=n>0 $ : $$\Gamma(1,x)=e^{-x}$$ $$\Gamma(2,x)=e^{-x}(1+x)$$ $$\Gamma(3,x)=e^{-x}(2+2x+x^2)$$