Hankel functions

Remarks

Both functions solve Bessel's equation $z^2 y'' + z y' + (z^2 - \nu^2) y = 0$, since they are linear combinations of $J_\nu$ and $Y_\nu$. They are independent: their Wronskian is $W(H^{(1)}_\nu, H^{(2)}_\nu)(z) = -\frac{4i}{\pi z}$. Example: for $\nu = \frac{1}{2}$ and $x > 0$, $J_{1/2}(x) = \sqrt{\frac{2}{\pi x}} \sin x$ and $Y_{1/2}(x) = -\sqrt{\frac{2}{\pi x}} \cos x$, so $H^{(1)}_{1/2}(x) = -i \sqrt{\frac{2}{\pi x}}\, e^{ix}$. For real $\nu$ and $x > 0$, $H^{(2)}_\nu(x)$ is the complex conjugate of $H^{(1)}_\nu(x)$. A non-example: $J_\nu + Y_\nu$ also solves Bessel's equation, but it is not a Hankel function, because the coefficient of $Y_\nu$ is not $\pm i$. They are also called Bessel functions of the third kind.