Hankel Functions, Bessel Functions of the Third Kind
Definition
A Hankel function, also known as a Bessel function of the third kind, is defined as the following two linear combinations of the Bessel function of the first kind $J_{\nu}$ and the Bessel function of the second kind $N_{\nu}$.
$$ H_{\nu}^{(1)}(x) = J_{\nu}(x)+iN_{\nu}(x) $$ $$ H_{\nu}^{(2)}(x) = J_{\nu}(x)-iN_{\nu}(x) $$
Explanation
It was introduced by the German mathematician Hermann Hankel in 1869. Specifically, $H_{\nu}^{(1)}$ is called the Hankel function of the first kind, and $H_{\nu}^{(2)}$ is called the Hankel function of the second kind.
To understand the definition, consider the differential equation $y^{\prime \prime}+y=0$. The solutions to this differential equation are $\cos x$ and $\sin x$. The general solution is represented as a linear combination of these, with the most commonly used form being $\cos x + \pm i \sin x=e^{\pm ix}$. Similarly, the general solution of the Bessel equation represented as a linear combination of two solutions, $J_{\nu}(x)$ and $N_{\nu}(x)$, is the Hankel function.