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1: 9.14 Incomplete Airy Functions
Incomplete Airy functions are defined by the contour integral (9.5.4) when one of the integration limits is replaced by a variable real or complex parameter. …
2: 33.18 Limiting Forms for Large
§33.18 Limiting Forms for Large
3: 4.31 Special Values and Limits
§4.31 Special Values and Limits
Table 4.31.1: Hyperbolic functions: values at multiples of 1 2 π i .
z 0 1 2 π i π i 3 2 π i
4.31.1 lim z 0 sinh z z = 1 ,
4.31.2 lim z 0 tanh z z = 1 ,
4.31.3 lim z 0 cosh z - 1 z 2 = 1 2 .
4: 4.4 Special Values and Limits
§4.4 Special Values and Limits
§4.4(iii) Limits
4.4.14 lim x 0 x a ln x = 0 , a > 0 ,
4.4.15 lim x x a e - x = 0 ,
4.4.18 lim n ( 1 + 1 n ) n = e .
5: 29.5 Special Cases and Limiting Forms
§29.5 Special Cases and Limiting Forms
29.5.4 lim k 1 - a ν m ( k 2 ) = lim k 1 - b ν m + 1 ( k 2 ) = ν ( ν + 1 ) - μ 2 ,
29.5.5 lim k 1 - Ec ν m ( z , k 2 ) Ec ν m ( 0 , k 2 ) = lim k 1 - Es ν m + 1 ( z , k 2 ) Es ν m + 1 ( 0 , k 2 ) = 1 ( cosh z ) μ F ( 1 2 μ - 1 2 ν , 1 2 μ + 1 2 ν + 1 2 1 2 ; tanh 2 z ) , m even,
lim Ec ν m ( z , k 2 ) = ce m ( 1 2 π - z , θ ) ,
lim Es ν m ( z , k 2 ) = se m ( 1 2 π - z , θ ) ,
6: 4.17 Special Values and Limits
§4.17 Special Values and Limits
Table 4.17.1: Trigonometric functions: values at multiples of 1 12 π .
θ sin θ cos θ tan θ csc θ sec θ cot θ
4.17.1 lim z 0 sin z z = 1 ,
4.17.2 lim z 0 tan z z = 1 .
4.17.3 lim z 0 1 - cos z z 2 = 1 2 .
7: 32.14 Combinatorics
32.14.1 lim N Prob ( N ( π ) - 2 N N 1 / 6 s ) = F ( s ) ,
The distribution function F ( s ) given by (32.14.2) arises in random matrix theory where it gives the limiting distribution for the normalized largest eigenvalue in the Gaussian Unitary Ensemble of n × n Hermitian matrices; see Tracy and Widom (1994). …
8: Preface
Thus the utilitarian value of the Handbook will be extended far beyond its original scope and the traditional limitations of printed media. …
9: 26.5 Lattice Paths: Catalan Numbers
§26.5(iv) Limiting Forms
26.5.7 lim n C ( n + 1 ) C ( n ) = 4 .
10: 3.9 Acceleration of Convergence
A transformation of a convergent sequence { s n } with limit σ into a sequence { t n } is called limit-preserving if { t n } converges to the same limit σ . The transformation is accelerating if it is limit-preserving and if …Similarly for convergent series if we regard the sum as the limit of the sequence of partial sums. … It may even fail altogether by not being limit-preserving. …