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11: 1.2 Elementary Algebra
1.2.40 𝐮 , 𝐯 = i = 1 n u i v i ¯ = 𝐯 H 𝐮 .
1.2.59 det ( 𝐀 ) = σ 𝔖 n sign σ i = 1 n a i , σ ( i ) .
12: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
1.18.9 v , w = n = 0 c n d n ¯ .
1.18.17 f ( x ) = n = 0 f , ϕ n ϕ n ( x ) = a b K ( x , y ) f ( y ) d y ,
1.18.66 ( z T ) 1 f , f = 𝝈 p | f ^ ( λ n ) | 2 z λ n + 𝝈 c | f ^ ( λ ) | 2 d λ z λ , f L 2 ( X ) , z 𝝈 .
13: Mathematical Introduction
complex plane (excluding infinity).
empty sums zero.
empty products unity.
14: 27.4 Euler Products and Dirichlet Series
27.4.1 n = 1 f ( n ) = p ( 1 + r = 1 f ( p r ) ) ,
27.4.2 n = 1 f ( n ) = p ( 1 f ( p ) ) 1 .
27.4.3 ζ ( s ) = n = 1 n s = p ( 1 p s ) 1 , s > 1 .
15: 27.6 Divisor Sums
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. …
16: Bibliography C
  • J. Chen (1966) On the representation of a large even integer as the sum of a prime and the product of at most two primes. Kexue Tongbao (Foreign Lang. Ed.) 17, pp. 385–386.
  • 17: 21.7 Riemann Surfaces
    21.7.17 P j U k = 1 4 θ [ 𝜶 k + 𝜼 1 ( P j ) 𝜷 k + 𝜼 2 ( P j ) ] ( 𝐳 k | 𝛀 ) = P j U c k = 1 4 θ [ 𝜶 k + 𝜼 1 ( P j ) 𝜷 k + 𝜼 2 ( P j ) ] ( 𝐳 k | 𝛀 ) .
    18: 21.6 Products
    21.6.3 j = 1 h θ ( k = 1 h T j k 𝐳 k | 𝛀 ) = 1 𝒟 g 𝐀 𝒦 𝐁 𝒦 e 2 π i tr [ 1 2 𝐀 T 𝛀 𝐀 + 𝐀 T [ 𝐙 + 𝐁 ] ] j = 1 h θ ( 𝐳 j + 𝛀 𝐚 j + 𝐛 j | 𝛀 ) ,
    21.6.4 j = 1 h θ [ k = 1 h T j k 𝐜 k k = 1 h T j k 𝐝 k ] ( k = 1 h T j k 𝐳 k | 𝛀 ) = 1 𝒟 g 𝐀 𝒦 𝐁 𝒦 e 2 π i j = 1 h 𝐛 j 𝐜 j j = 1 h θ [ 𝐚 j + 𝐜 j 𝐛 j + 𝐝 j ] ( 𝐳 j | 𝛀 ) ,
    19: 16.14 Partial Differential Equations
    In addition to the four Appell functions there are 24 other sums of double series that cannot be expressed as a product of two F 1 2 functions, and which satisfy pairs of linear partial differential equations of the second order. …
    20: 1.10 Functions of a Complex Variable
    The product n = 1 ( 1 + a n ) , with a n 1 for all n , converges iff n = 1 ln ( 1 + a n ) converges; and it converges absolutely iff n = 1 | a n | converges. …