The leading nonzero residual is then given entirely by the shifting. Since all the matrix elements are at indices which have a small distance from the large index position (m, m), it helps to introduce two temporary notations: A [ r, k ] = A (m + r)(m + k) for the matrices, and (dA / dJ)[ r ] for the r'th Fourier components of classical quantities, ( A B − − B A ) [ 0 , k ] = ∑ ∑ r = − − ∞ ∞ ∞ ∞ ( A [ 0 , r ] B [ r , k ] − − A [ r , k ] B [ 0 , r ] ) = ∑ ∑ r ( A [ − − r + k , k ] + ( r − − k ) d A d J [ r ] ) ( B [ 0 , k − − r ] + r d B d J [ r − − k ] ) − − ∑ ∑ r A [ r , k ] B [ 0 , r ] .