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of the second kind

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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 10.26.7: I ~ 1 / 2 ( x ) , K ~ 1 / 2 ( x ) , 0.01 x 3 . 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: 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). …
3: 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 ) ¯ .
4: 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 )
5: 26.8 Set Partitions: Stirling Numbers
S ( n , k ) denotes the Stirling number of the second kind: the number of partitions of { 1 , 2 , , n } into exactly k nonempty subsets. …
Table 26.8.2: Stirling numbers of the second kind S ( n , k ) .
n k
S ( n , 0 ) = 0 ,
S ( n , 1 ) = 1 ,
26.8.22 S ( n , k ) = k S ( n - 1 , k ) + S ( n - 1 , k - 1 ) ,
6: 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 ) | .
7: 10.42 Zeros
Properties of the zeros of I ν ( z ) and K ν ( z ) may be deduced from those of J ν ( z ) and H ν ( 1 ) ( z ) , respectively, by application of the transformations (10.27.6) and (10.27.8). … The distribution of the zeros of K n ( n z ) in the sector - 3 2 π ph z 1 2 π in the cases n = 1 , 5 , 10 is obtained on rotating Figures 10.21.2, 10.21.4, 10.21.6, respectively, through an angle - 1 2 π so that in each case the cut lies along the positive imaginary axis. … K n ( z ) has no zeros in the sector | ph z | 1 2 π ; this result remains true when n is replaced by any real number ν . For the number of zeros of K ν ( z ) in the sector | ph z | π , when ν is real, see Watson (1944, pp. 511–513). For z -zeros of K ν ( z ) , with complex ν , see Ferreira and Sesma (2008). …
8: 10.28 Wronskians and Cross-Products
10.28.2 𝒲 { K ν ( z ) , I ν ( z ) } = I ν ( z ) K ν + 1 ( z ) + I ν + 1 ( z ) K ν ( z ) = 1 / z .
9: 26.1 Special Notation
x real variable.
( m n ) binomial coefficient.
S ( n , k ) Stirling numbers of the second kind.
Other notations for S ( n , k ) , the Stirling numbers of the second kind, include 𝒮 n ( k ) (Fort (1948)), 𝔖 n k (Jordan (1939)), σ n k (Moser and Wyman (1958b)), ( n k ) B n - k ( - k ) (Milne-Thomson (1933)), S 2 ( k , n - k ) (Carlitz (1960), Gould (1960)), { n k } (Knuth (1992), Graham et al. (1994), Rosen et al. (2000)), and also an unconventional symbol in Abramowitz and Stegun (1964, Chapter 24).
10: 19.4 Derivatives and Differential Equations
d E ( k ) d k = E ( k ) - K ( k ) k ,
d ( E ( k ) - K ( k ) ) d k = - k E ( k ) k 2 ,
If ϕ = π / 2 , then these two equations become hypergeometric differential equations (15.10.1) for K ( k ) and E ( k ) . …