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1: 7.8 Inequalities
§7.8 Inequalities
Let M ( x ) denote Mills’ ratio: …
7.8.2 1 x + x 2 + 2 < M ( x ) 1 x + x 2 + ( 4 / π ) , x 0 ,
7.8.3 π 2 π x + 2 M ( x ) < 1 x + 1 , x 0 ,
7.8.4 M ( x ) < 2 3 x + x 2 + 4 , x > - 1 2 2 ,
2: 12.6 Continued Fraction
For a continued-fraction expansion of the ratio U ( a , x ) / U ( a - 1 , x ) see Cuyt et al. (2008, pp. 340–341).
3: 4.22 Infinite Products and Partial Fractions
4.22.1 sin z = z n = 1 ( 1 - z 2 n 2 π 2 ) ,
4.22.2 cos z = n = 1 ( 1 - 4 z 2 ( 2 n - 1 ) 2 π 2 ) .
4.22.3 cot z = 1 z + 2 z n = 1 1 z 2 - n 2 π 2 ,
4.22.4 csc 2 z = n = - 1 ( z - n π ) 2 ,
4.22.5 csc z = 1 z + 2 z n = 1 ( - 1 ) n z 2 - n 2 π 2 .
4: 4.36 Infinite Products and Partial Fractions
4.36.1 sinh z = z n = 1 ( 1 + z 2 n 2 π 2 ) ,
4.36.2 cosh z = n = 1 ( 1 + 4 z 2 ( 2 n - 1 ) 2 π 2 ) .
4.36.3 coth z = 1 z + 2 z n = 1 1 z 2 + n 2 π 2 ,
4.36.4 csch 2 z = n = - 1 ( z - n π i ) 2 ,
4.36.5 csch z = 1 z + 2 z n = 1 ( - 1 ) n z 2 + n 2 π 2 .
5: 3.12 Mathematical Constants
3.12.1 π = 3.14159 26535 89793 23846
3.12.2 π = 4 0 1 d t 1 + t 2 .
6: 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 ) .
7: 21.8 Abelian Functions
For every Abelian function, there is a positive integer n , such that the Abelian function can be expressed as a ratio of linear combinations of products with n factors of Riemann theta functions with characteristics that share a common period lattice. …
8: 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 ) ,
22.12.8 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 ) ,
9: 28.3 Graphics
10: 7.9 Continued Fractions
7.9.1 π e z 2 erfc z = z z 2 + 1 2 1 + 1 z 2 + 3 2 1 + 2 z 2 + , z > 0 ,
7.9.2 π e z 2 erfc z = 2 z 2 z 2 + 1 - 1 2 2 z 2 + 5 - 3 4 2 z 2 + 9 - , z > 0 ,
7.9.3 w ( z ) = i π 1 z - 1 2 z - 1 z - 3 2 z - 2 z - , z > 0 .