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of the first and second kinds

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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
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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
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Figure 10.26.7: I ~ 1 / 2 ( x ) , K ~ 1 / 2 ( x ) , 0.01 x 3 . Magnify
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Figure 10.26.8: I ~ 1 ( x ) , K ~ 1 ( x ) , 0.01 x 3 . Magnify
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Figure 10.26.9: I ~ 5 ( x ) , K ~ 5 ( x ) , 0.01 x 3 . Magnify
2: 10.45 Functions of Imaginary Order
10.45.3 I ~ ν ( x ) = I ~ ν ( x ) , K ~ ν ( x ) = K ~ ν ( x ) ,
and I ~ ν ( x ) , K ~ ν ( x ) are real and linearly independent solutions of (10.45.1): … 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 ) .
10.34.3 I ν ( z e m π i ) = ( i / π ) ( ± e m ν π i K ν ( z e ± π i ) e ( m 1 ) ν π i 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 ) ,
4: 26.8 Set Partitions: Stirling Numbers
§26.8(i) Definitions
Table 26.8.1: Stirling numbers of the first kind s ( n , k ) .
n k
Table 26.8.2: Stirling numbers of the second kind S ( n , k ) .
n k
§26.8(ii) Generating Functions
§26.8(v) Identities
5: 19.4 Derivatives and Differential Equations
d ( E ( k ) k 2 K ( k ) ) d k = k K ( k ) ,
d E ( k ) d k = E ( k ) K ( k ) k ,
d ( E ( k ) K ( k ) ) d k = k E ( k ) k 2 ,
19.4.3 d 2 E ( k ) d k 2 = 1 k d K ( k ) d k = k 2 K ( k ) E ( k ) k 2 k 2 ,
If ϕ = π / 2 , then these two equations become hypergeometric differential equations (15.10.1) for K ( k ) and E ( k ) . …
6: 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 .
7: 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 . …
8: 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). …
9: 26.1 Special Notation
x real variable.
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. …