Completeness of Vector Spherical Wave Functions

An approximate solution to the null-field equations can be obtained by approximating the surface fields e-, and h, by the complete set of regular vector spherical wave functions for the interior domain (or the interior wave... 

In the present analysis we will derive the expression of the transition matrix by using the translation properties of the vector spherical wave functions. The completeness property of the vector spherical wave functions on two enclosing surfaces, which is essential in our analysis, is established in Appendix D. 

The surface fields ei i and hi i are the tangential components of the electric and magnetic fields in the domain bounded by the closed surfaces Sx and 52. Taking into account the completeness property of the system of regular and radiating vector spherical wave functions on two enclosing surfaces... 

Returning to the scattering problem of a particle near an arbitrary infinite surface, we see that our analysis is complete if, according to Step 1, we are able to represent the plane electromagnetic wave and the radiating vector spherical wave functions and as integrals over vector plane waves. 

In the same manner we can prove the completeness and linear independence of the system of distributed vector spherical wave functions and... 

This volume is intended for engineering and physics students as well as researchers in scattering theory, and therefore we decided to leave out rigorous mathematical details. The properties of scalar and vector spherical wave functions, addition theorems under translation and rotation of the coordinate systems and some completeness results are presented in appendices. These can be regarded as a collection of necessary formulas. 

K.A. Aydin, A. Hizal, On the completeness of the spherical vector wave functions, J. Math. Anal. Appl. 117, 428 (1986)... 

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