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1: 19.38 Approximations
Minimax polynomial approximations (§3.11(i)) for K ( k ) and E ( k ) in terms of m = k 2 with 0 m < 1 can be found in Abramowitz and Stegun (1964, §17.3) with maximum absolute errors ranging from 4×10⁻⁵ to 2×10⁻⁸. Approximations of the same type for K ( k ) and E ( k ) for 0 < k 1 are given in Cody (1965a) with maximum absolute errors ranging from 4×10⁻⁵ to 4×10⁻¹⁸. …
2: 3.1 Arithmetics and Error Measures
The lower and upper bounds for the absolute values of the nonzero machine numbers are given by … Also in this arithmetic generalized precision can be defined, which includes absolute error and relative precision (§3.1(v)) as special cases. … If x is an approximation to a real or complex number x , then the absolute error is
3.1.8 ϵ a = | x x | .
3.1.9 ϵ r = | x x x | = ϵ a | x | .
3: 6.13 Zeros
Ci ( x ) and si ( x ) each have an infinite number of positive real zeros, which are denoted by c k , s k , respectively, arranged in ascending order of absolute value for k = 0 , 1 , 2 , . …
4: 23.16 Graphics
In Figures 23.16.2 and 23.16.3, height corresponds to the absolute value of the function and color to the phase. …
5: 1.9 Calculus of a Complex Variable
1.9.12 | z ¯ | = | z | ,
§1.9(v) Infinite Sequences and Series
1.9.47 | f n ( z ) f ( z ) | < ϵ
For z in | z z 0 | ρ ( < R ), the convergence is absolute and uniform. …
6: 1.10 Functions of a Complex Variable
1.10.2 e z = 1 + z 1 ! + z 2 2 ! + , | z | < ,
§1.10(ix) Infinite Products
The convergence of the product is absolute if n = 1 ( 1 + | a n | ) converges. …
1.10.20 | ln ( 1 + a n ( z ) ) | M n , n N ,
1.10.26 F ( x ; z ) = n = 0 p n ( x ) z n , | z | < R .
7: 1.4 Calculus of One Variable
1.4.2 | f ( c + α ) f ( c ) | < ϵ ,
Absolute convergence also implies convergence. … In particular, absolute continuity occurs if the function α ( x ) is differentiable, α ( x ) = w ( x ) with w ( x ) continuous. …
8: 1.15 Summability Methods
1.15.28 n = i ( sign n ) F ( n ) r | n | e i n θ ;
1.15.30 lim ϵ 0 + e ϵ | t | f ( t ) d t = L .
1.15.32 lim R R R ( 1 | t | R ) f ( t ) d t = L .
1.15.35 | x | δ P ( x , y ) d x 0 , as y 0 .
1.15.38 lim y 0 + | h ( x , y ) f ( x ) | d x = 0 .
9: 4.3 Graphics
In the graphics shown in this subsection height corresponds to the absolute value of the function and color to the phase. …
10: 5.3 Graphics
In the graphics shown in this subsection, both the height and color correspond to the absolute value of the function. …