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# Table of Properties of Vector Harmonics

1. Basic Definitions &ell#ell;&ell#ell;m & = &1&ell#ell;(&ell#ell;+1) Y_&ell#ell;,m
&ell#ell;&ell#ell;-1m & = &-1&ell#ell;(2&ell#ell;+1) [ -&ell#ell; + i×]Y_&ell#ell;,m
&ell#ell;&ell#ell;+1m & = & -1(&ell#ell;+1)(2&ell#ell;+1) [ (&ell#ell;+1) + i×]Y_&ell#ell;,m

2. Eigenvalues ( are integral): J^2 j &ell#ell;m & = & j(j+1)j &ell#ell;m
L^2 j &ell#ell;m & = & &ell#ell;(&ell#ell;+1)j &ell#ell;m
J_z j &ell#ell;m & = & m j &ell#ell;m

3. Projective Orthonormality:

4. Complex Conjugation:

5. Addition Theorem (LCB notes corrupt - this needs to be checked): j &ell#ell;m &ast#ast;·j' &ell#ell;'m' & = & &sum#sum;_n (-1)^m+1(2&ell#ell;+1)(2&ell#ell;'+1)(2j'+1)(2j+1)4&pi#pi;(2n+1) ×
& &         C_000^&ell#ell;&ell#ell;'nC_0,-m,m'^jj'n W(j&ell#ell;j'&ell#ell;';n) Y_n,(m'-m)

6. For any function of only: (&ell#ell;&ell#ell;mF) & = & 0
(&ell#ell;&ell#ell;-1mF) & = & &ell#ell;2&ell#ell;+1 [ (&ell#ell;-1)Fr - dFdr] Y_&ell#ell;,m
(&ell#ell;&ell#ell;+1mF) & = & &ell#ell;+12&ell#ell;+1 [ (&ell#ell;+2)Fr - dFdr] Y_&ell#ell;,m

7. Ditto: i(&ell#ell;&ell#ell;mF) & = & &ell#ell;+12&ell#ell;+1 [ (&ell#ell;+1)Fr + dFdr]&ell#ell;&ell#ell;-1m + &ell#ell;2&ell#ell;+1 [-&ell#ell;Fr + dFdr]&ell#ell;&ell#ell;+1m
i(&ell#ell;&ell#ell;-1mF) & = & -&ell#ell;+12&ell#ell;+1 [ (&ell#ell;-1)Fr - dFdr]&ell#ell;&ell#ell;m
i(&ell#ell;&ell#ell;+1mF) & = & &ell#ell;2&ell#ell;+1 [ (&ell#ell;+2)Fr - dFdr]&ell#ell;&ell#ell;m

8. This puts the VSHs into vector form:

9. Hansen Multipole Properties _L & = & 0
_L & = & 0
_L & = & i k f_&ell#ell;(kr) Y_L() _L & = & -ik _L
_L & = & ik _L
_L & = & 0

10. Hansen Multipole Explicit Forms _L & = & f_&ell#ell;(kr) &ell#ell;&ell#ell;m
_L & = & &ell#ell;+12 &ell#ell;+1 f_&ell#ell;- 1(kr) &ell#ell;, &ell#ell;-1m - &ell#ell;2 &ell#ell;+ 1 f_&ell#ell;+ 1(kr) &ell#ell;,&ell#ell;+1m
_L & = & &ell#ell;2 &ell#ell;+ 1 f_&ell#ell;- 1(kr) &ell#ell;, &ell#ell;-1m + &ell#ell;+12 &ell#ell;+1 f_&ell#ell;+ 1(kr) &ell#ell;,&ell#ell;+1m _L & = & f_&ell#ell;(kr) &ell#ell;&ell#ell;m
_L & = & 1kr { d   d(kr) (kr f_&ell#ell;) (i×&ell#ell;&ell#ell;m ) - &ell#ell;(&ell#ell;+ 1) f_&ell#ell;Y_L }
_L & = & &ell#ell;(&ell#ell;+ 1) 1kr (i ×f_&ell#ell;&ell#ell;&ell#ell;m) - [d   d(kr) f_&ell#ell; ] Y_L

Next: Optical Scattering Up: The Hansen Multipoles Previous: Concluding Remarks About Multipoles   Contents
Robert G. Brown 2017-07-11