Skip to main content

Questions related to Bessel functions.

Questions and problems related to cylindrical harmonics or Bessel functions, normally taken to satisfy the differential equation $$ x^2 y'' + x y' + (x^2-\nu^2)y = 0, \tag{1} $$ (Bessel's equation) or its modification $$ x^2 y'' + x y' + (x^2+\nu^2)y = 0. \tag{2} $$ The solutions to (1) are called $J_{\nu}$ and $Y_{\nu}$; those to (2) are called $I_{\nu}$ and $K_{\nu}$. Special complex combinations of $J_{\nu}$ and $Y_{\nu}$ are also called Hankel functions, $$ H_{\nu}^{(1)} = J_{\nu} + i Y_{\nu}, \qquad H_{\nu}^{(2)} = J_{\nu} - i Y_{\nu}. $$