A new Sommerfeld-Watson transform in 3D

We consider the problem of high-frequency wave scattering by a sphere. The exact solution in the form of the partial wave expansion, which is known as a Mie series, has been long established. The number of summation terms needed to get an accurate solution increases as the frequency becomes higher, making the computational cost very expensive. A traditional way to overcome this difficulty is to use the Sommerfeld-Watson (S-W) transformation to change the summation into a residue series. Although excellent research has been done on this topic, the results for the 3D case lack the physical clarity that is obviated in the 2D case. A new solution in terms of a residue series is introduced. In order to give a better physical interpretation, the function Q~(θ) has been used as the spherical travelling wave. A new residue series using an S-W transformation based on this has been derived which exhibits symmetry in its mathematical structure. Moreover, it offers the same physical clarity in its interpretation for the 3D case as the 2D case.