About the Project
NIST

second

AdvancedHelp

(0.001 seconds)

1—10 of 314 matching pages

1: 10.26 Graphics
See accompanying text
Figure 10.26.1: I 0 ( x ) , I 1 ( x ) , K 0 ( x ) , K 1 ( x ) , 0 x 3 . Magnify
See accompanying text
Figure 10.26.2: e - x I 0 ( x ) , e - x I 1 ( x ) , e x K 0 ( x ) , e x K 1 ( x ) , 0 x 10 . Magnify
See accompanying text
Figure 10.26.8: I ~ 1 ( x ) , K ~ 1 ( x ) , 0.01 x 3 . Magnify
See accompanying text
Figure 10.26.9: I ~ 5 ( x ) , K ~ 5 ( x ) , 0.01 x 3 . Magnify
See accompanying text
Figure 10.26.10: K ~ 5 ( x ) , 0.01 x 3 . Magnify
2: 26.21 Tables
§26.21 Tables
Abramowitz and Stegun (1964, Chapter 24) tabulates binomial coefficients ( m n ) for m up to 50 and n up to 25; extends Table 26.4.1 to n = 10 ; tabulates Stirling numbers of the first and second kinds, s ( n , k ) and S ( n , k ) , for n up to 25 and k up to n ; tabulates partitions p ( n ) and partitions into distinct parts p ( 𝒟 , n ) for n up to 500. … It also contains a table of Gaussian polynomials up to [ 12 6 ] q . …
3: 26.8 Set Partitions: Stirling Numbers
§26.8(i) Definitions
S ( n , k ) denotes the Stirling number of the second kind: the number of partitions of { 1 , 2 , , n } into exactly k nonempty subsets. …
§26.8(ii) Generating Functions
§26.8(iv) Recurrence Relations
§26.8(v) Identities
4: 10.34 Analytic Continuation
10.34.2 K ν ( z e m π i ) = e - m ν π i K ν ( z ) - π i sin ( m ν π ) csc ( ν π ) I ν ( z ) .
10.34.4 K ν ( z e m π i ) = csc ( ν π ) ( ± sin ( m ν π ) K ν ( z e ± π i ) sin ( ( m 1 ) ν π ) K ν ( z ) ) .
10.34.5 K n ( z e m π i ) = ( - 1 ) m n K n ( z ) + ( - 1 ) n ( m - 1 ) - 1 m π i I n ( z ) ,
10.34.6 K n ( z e m π i ) = ± ( - 1 ) n ( m - 1 ) m K n ( z e ± π i ) ( - 1 ) n m ( m 1 ) K n ( z ) .
K ν ( z ¯ ) = K ν ( z ) ¯ .
5: 10.45 Functions of Imaginary Order
and I ~ ν ( x ) , K ~ ν ( x ) are real and linearly independent solutions of (10.45.1): … The corresponding result for K ~ ν ( x ) is given by … In consequence of (10.45.5)–(10.45.7), I ~ ν ( x ) and K ~ ν ( x ) comprise a numerically satisfactory pair of solutions of (10.45.1) when x is large, and either I ~ ν ( x ) and ( 1 / π ) sinh ( π ν ) K ~ ν ( x ) , or I ~ ν ( x ) and K ~ ν ( x ) , comprise a numerically satisfactory pair when x is small, depending whether ν 0 or ν = 0 . … For graphs of I ~ ν ( x ) and K ~ ν ( x ) see §10.26(iii). For properties of I ~ ν ( x ) and K ~ ν ( x ) , including uniform asymptotic expansions for large ν and zeros, see Dunster (1990a). …
6: 26.17 The Twelvefold Way
In this table ( k ) n is Pochhammer’s symbol, and S ( n , k ) and p k ( n ) are defined in §§26.8(i) and 26.9(i). …
Table 26.17.1: The twelvefold way.
elements of N elements of K f unrestricted f one-to-one f onto
labeled labeled k n ( k - n + 1 ) n k ! S ( n , k )
labeled unlabeled S ( n , 1 ) + S ( n , 2 ) + + S ( n , k ) { 1 n k 0 n > k S ( n , k )
7: 9.15 Mathematical Applications
Airy functions play an indispensable role in the construction of uniform asymptotic expansions for contour integrals with coalescing saddle points, and for solutions of linear second-order ordinary differential equations with a simple turning point. …
8: 30.5 Functions of the Second Kind
§30.5 Functions of the Second Kind
30.5.1 Qs n m ( x , γ 2 ) , n = m , m + 1 , m + 2 , .
30.5.2 Qs n m ( - x , γ 2 ) = ( - 1 ) n - m + 1 Qs n m ( x , γ 2 ) ,
30.5.4 𝒲 { Ps n m ( x , γ 2 ) , Qs n m ( x , γ 2 ) } = ( n + m ) ! ( 1 - x 2 ) ( n - m ) ! A n m ( γ 2 ) A n - m ( γ 2 ) ( 0 ) ,
9: 28.5 Second Solutions fe n , ge n
Second solutions of (28.2.1) are given by … …
Odd Second Solutions
Even Second Solutions
10: 10.37 Inequalities; Monotonicity
If ν ( 0 ) is fixed, then throughout the interval 0 < x < , I ν ( x ) is positive and increasing, and K ν ( x ) is positive and decreasing. If x ( > 0 ) is fixed, then throughout the interval 0 < ν < , I ν ( x ) is decreasing, and K ν ( x ) is increasing. …
10.37.1 | K ν ( z ) | < | K μ ( z ) | .