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1: 1.1 Special Notation
x , y real variables.
f , g inner, or scalar, product for real or complex vectors or functions.
𝐀 1 inverse of the square matrix 𝐀
det ( 𝐀 ) determinant of the square matrix 𝐀
tr ( 𝐀 ) trace of the square matrix 𝐀
𝐀 adjoint of the square matrix 𝐀
2: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
1.18.11 a b | f ( x ) | 2 d α ( x ) < .
Functions f , g L 2 ( X , d α ) for which f g , f g = 0 are identified with each other. The space L 2 ( X , d α ) becomes a separable Hilbert space with inner product
3: 1.2 Elementary Algebra
§1.2(vi) Square Matrices
Special Forms of Square Matrices
Norms of Square Matrices
Non-Defective Square Matrices
4: 26.12 Plane Partitions
26.12.9 ( h = 1 r j = 1 s h + j + t 1 h + j 1 ) 2 ;
26.12.10 ( h = 1 r j = 1 s h + j + t 1 h + j 1 ) ( h = 1 r + 1 j = 1 s h + j + t 1 h + j 1 ) ;
26.12.11 ( h = 1 r + 1 j = 1 s h + j + t 1 h + j 1 ) ( h = 1 r j = 1 s + 1 h + j + t 1 h + j 1 ) .
26.12.13 h = 1 r j = 1 r h + j + t 1 h + j 1 ;
where σ 2 ( j ) is the sum of the squares of the divisors of j . …
5: 20.12 Mathematical Applications
For applications of θ 3 ( 0 , q ) to problems involving sums of squares of integers see §27.13(iv), and for extensions see Estermann (1959), Serre (1973, pp. 106–109), Koblitz (1993, pp. 176–177), and McKean and Moll (1999, pp. 142–143). For applications of Jacobi’s triple product (20.5.9) to Ramanujan’s τ ( n ) function and Euler’s pentagonal numbers see Hardy and Wright (1979, pp. 132–160) and McKean and Moll (1999, pp. 143–145). …
6: 27.6 Divisor Sums
27.6.1 d | n λ ( d ) = { 1 , n  is a square , 0 , otherwise .
27.6.2 d | n μ ( d ) f ( d ) = p | n ( 1 f ( p ) ) , n > 1 .
Generating functions, Euler products, and Möbius inversion are used to evaluate many sums extended over divisors. …
7: 1.3 Determinants, Linear Operators, and Spectral Expansions
The determinant of an upper or lower triangular, or diagonal, square matrix 𝐀 is the product of the diagonal elements det ( 𝐀 ) = i = 1 n a i i . …
8: 4.35 Identities
§4.35(ii) Squares and Products
9: 4.21 Identities
§4.21(ii) Squares and Products
10: 18.39 Applications in the Physical Sciences
All are written in the same form as the product of three factors: the square root of a weight function w ( x ) , the corresponding OP or EOP, and constant factors ensuring unit normalization. …