M. Minnaert's The Nature of Light and Colour in the Open Air PDF
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Extra resources for The Nature of Light and Colour in the Open Air
Sample text
25) over [−π, π ]. 12, the angular average of the left hand side becomes S− √ −1 I aα . 72), it follows that r(π ) = r(−π ). 9 implies that π −π aα−β (φ)dφ = (−1)α−β−1 This proves the theorem. r(φ)α−β α−β φ=π φ=−π = 0, β = 1, 2, . . , α − 1.
First, we consider the case p2 = p3 . 119) where AC − F 2 − 2F L , 2AL √ √ −(AC − F 2 )( AC + F + 2L)( AC − F − 2L) . 101) and = cos α, is also equivalent to √ = 1 or 3 = −1 or AC − F − 2L = 0. 120) Suppose that 3 = 1 or 3 = −1. Then we have sin α = 0, cos α = 1 and p1 = p2 = √ p3 = −1. 109) is one at p = −1. 29) has√exactly two linearly independent eigenvectors associated with the eigenvalue p = −1 of multiplicity 3. 29) is degenerate and belongs to Case D1. 74) we obtain 0 0 Z 11 = Z 22 = 2A(A − N) , 3A − N √ (A − N)2 0 = − −1 Z 12 3A − N 3 0 = L, Z 33 0 = −Z 21 , 0 0 0 0 Z 13 = Z 23 = Z 31 = Z 32 = 0.
121), we obtain 1 u(y1 , y2 ) = √ 2π 1 = √ 2π 6 α=1 R e 6 α=1 R e √ −1 ξ1 ( pα y2 +y1 ) cα (ξ1 ) aα dξ1 √ −1 ξ1 (m·x+ pα n·x) cα (ξ1 ) aα dξ1 . 135) α=1 and becomes for ξ1 > 0, 6 cα (ξ1 ) aα e √ −1 ξ1 (m·x+ pα n·x) . 45). 4). 48). Let α (α = 1, 2) be linearly independent eigenvectors of N associated with lα a the eigenvalues pα (α = 1, 2, Im pα > 0), and let 3 be a generalized eigenvector l3 which satisfies N a a3 − p2 3 l3 l3 = a2 . 44). 137) be the projection operators on the eigenspaces of N (α = 1, 4) or on the generalized eigenspaces of N (α = 2, 5) associated with pα .