Part 1 : Nanoparticles Module 6 : Mie theory for Scattering and absorption by                    a sphere and ellipsoids
Lecture 1 : Helmholtz equation for fields and Vector harmonics
 
The linearly independent solutions for $Z$ are the Bessel functions of the first kind $J_\nu$ and second kind $Y_\nu$ (sometimes denoted by $N_\nu$), where $\nu=l+1/2$, the order is half integral. Therefore the linearly independent solutions for $R$ are the spherical Bessel functions,
\begin{displaymath}j_l(\rho)=\sqrt{\frac{\pi}{2\rho}}\,J_{l+\frac{1}{2}}\,\quad\... ...d\quad y_l(\rho)=\sqrt{\frac{\pi}{2\rho}}\,Y_{l+\frac{1}{2}}\,,\end{displaymath}
where the constant factor $\sqrt{\pi/2}$ is introduced to comply with standard definitions. Two linearly independent solutions which can be constructed by taking suitable linear combinations of $j_l$ and $y_l$ are the spherical Bessel functions of third kind (also known as spherical Hankel functions) :
\begin{displaymath}h_l^{(1)}(\rho)=j_l(\rho)+iy_l(\rho)\,,\quad\quad\quad{\rm and}\quad\quad h_l^{(2)}(\rho)=j_l(\rho)-iy_l(\rho)\,,\end{displaymath}
which are relevant for scattering problem, as will be seen later.