Commun. Comput. Phys., Notice: Undefined index: year in /var/www/html/issue/abstract/readabs.php on line 20 Notice: Undefined index: ppage in /var/www/html/issue/abstract/readabs.php on line 21 Notice: Undefined index: issue in /var/www/html/issue/abstract/readabs.php on line 23 Volume 4. Diagonalizations of Vector and Tensor Addition Theorems B. He 1, W. C. Chew 2*1 Department of Electrical and Computer Engineering, University of Illinois, Urbana-Champaign, Urbana, IL 61801-2918, USA. 2 Department of Electrical and Computer Engineering, University of Illinois, Urbana-Champaign, Urbana, IL 61801-2918, USA; and Faculty of Engineering, The University of Hong Kong, Hong Kong. Received 4 September 2007; Accepted (in revised version) 17 March 2008 Available online 16 May 2008 Abstract Based on the generalizations of the Funk-Hecke formula and the Rayleigh plan-wave expansion formula, an alternative and succinct derivation of the addition theorem for general tensor field is obtained. This new derivation facilitates the diagonalization of the tensor addition theorem. In order to complete this derivation, we have carried out the evaluation of the generalization of the Gaunt coefficient for tensor fields. Since vector fields (special case of tensor fields) are very useful in practice, we discuss vector multipole fields and vector addition theorem in details. The work is important in multiple scattering and fast algorithms in wave physics. AMS subject classifications: 43A90, 78A45, 78A25, 78M15 PACS: 02.70.Pt., 03.50.De, 89.20.Ff Key words: Addition theorem, vector field, tensor field, fast algorithm. *Corresponding author. Email: bohe@uiuc.edu (B. He), wcchew@hku.hk, w-chew@uiuc.edu (W. C. Chew)