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1: 6.15 Sums
§6.15 Sums
6.15.2 n = 1 si ( π n ) n = 1 2 π ( ln π 1 ) ,
6.15.3 n = 1 ( 1 ) n Ci ( 2 π n ) = 1 ln 2 1 2 γ ,
6.15.4 n = 1 ( 1 ) n si ( 2 π n ) n = π ( 3 2 ln 2 1 ) .
2: 16.20 Integrals and Series
§16.20 Integrals and Series
Integrals of the Meijer G -function are given in Apelblat (1983, §19), Erdélyi et al. (1953a, §5.5.2), Erdélyi et al. (1954a, §§6.9 and 7.5), Luke (1969a, §3.6), Luke (1975, §5.6), Mathai (1993, §3.10), and Prudnikov et al. (1990, §2.24). …
3: 1.7 Inequalities
Cauchy–Schwarz Inequality
Hölder’s Inequality
Minkowski’s Inequality
Cauchy–Schwarz Inequality
Minkowski’s Inequality
4: 6.6 Power Series
6.6.1 Ei ( x ) = γ + ln x + n = 1 x n n ! n , x > 0 .
6.6.2 E 1 ( z ) = γ ln z n = 1 ( 1 ) n z n n ! n .
6.6.4 Ein ( z ) = n = 1 ( 1 ) n 1 z n n ! n ,
6.6.5 Si ( z ) = n = 0 ( 1 ) n z 2 n + 1 ( 2 n + 1 ) ! ( 2 n + 1 ) ,
5: 22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series
22.12.2 2 K k sn ( 2 K t , k ) = n = π sin ( π ( t ( n + 1 2 ) τ ) ) = n = ( m = ( 1 ) m t m ( n + 1 2 ) τ ) ,
22.12.3 2 i K k cn ( 2 K t , k ) = n = ( 1 ) n π sin ( π ( t ( n + 1 2 ) τ ) ) = n = ( m = ( 1 ) m + n t m ( n + 1 2 ) τ ) ,
22.12.8 2 K dc ( 2 K t , k ) = n = π sin ( π ( t + 1 2 n τ ) ) = n = ( m = ( 1 ) m t + 1 2 m n τ ) ,
22.12.11 2 K ns ( 2 K t , k ) = n = π sin ( π ( t n τ ) ) = n = ( m = ( 1 ) m t m n τ ) ,
22.12.12 2 K ds ( 2 K t , k ) = n = ( 1 ) n π sin ( π ( t n τ ) ) = n = ( m = ( 1 ) m + n t m n τ ) ,
6: 7.18 Repeated Integrals of the Complementary Error Function
7.18.6 i n erfc ( z ) = k = 0 ( 1 ) k z k 2 n k k ! Γ ( 1 + 1 2 ( n k ) ) .
7.18.14 i n erfc ( z ) 2 π e z 2 ( 2 z ) n + 1 m = 0 ( 1 ) m ( 2 m + n ) ! n ! m ! ( 2 z ) 2 m , z , | ph z | 3 4 π δ ( < 3 4 π ) .
7: 19.21 Connection Formulas
19.21.11 6 R G ( x , y , z ) = 3 ( x + y + z ) R F ( x , y , z ) x 2 R D ( y , z , x ) = x ( y + z ) R D ( y , z , x ) ,
8: 25.16 Mathematical Applications
25.16.6 H ( s ) = ζ ( s ) + γ ζ ( s ) + 1 2 ζ ( s + 1 ) + r = 1 k ζ ( 1 2 r ) ζ ( s + 2 r ) + n = 1 1 n s n B ~ 2 k + 1 ( x ) x 2 k + 2 d x ,
25.16.7 H ( s ) = 1 2 ζ ( s + 1 ) + ζ ( s ) s 1 r = 1 k ( s + 2 r 2 2 r 1 ) ζ ( 1 2 r ) ζ ( s + 2 r ) ( s + 2 k 2 k + 1 ) n = 1 1 n n B ~ 2 k + 1 ( x ) x s + 2 k + 1 d x .
9: 36.10 Differential Equations
36.10.2 K + 1 Ψ K ( 𝐱 ) x 1 K + 1 + m = 1 K ( i ) m K 2 ( m x m K + 2 ) m 1 Ψ K ( 𝐱 ) x 1 m 1 = 0 .
10: 22.11 Fourier and Hyperbolic Series
22.11.3 dn ( z , k ) = π 2 K + 2 π K n = 1 q n cos ( 2 n ζ ) 1 + q 2 n .
22.11.6 nd ( z , k ) = π 2 K k + 2 π K k n = 1 ( 1 ) n q n cos ( 2 n ζ ) 1 + q 2 n .
22.11.7 ns ( z , k ) π 2 K csc ζ = 2 π K n = 0 q 2 n + 1 sin ( ( 2 n + 1 ) ζ ) 1 q 2 n + 1 ,
22.11.13 sn 2 ( z , k ) = 1 k 2 ( 1 E K ) 2 π 2 k 2 K 2 n = 1 n q n 1 q 2 n cos ( 2 n ζ ) .