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1: 12 Parabolic Cylinder Functions
Chapter 12 Parabolic Cylinder Functions
2: 34.6 Definition: 9 j Symbol
34.6.1 { j 11 j 12 j 13 j 21 j 22 j 23 j 31 j 32 j 33 } = all  m r s ( j 11 j 12 j 13 m 11 m 12 m 13 ) ( j 21 j 22 j 23 m 21 m 22 m 23 ) ( j 31 j 32 j 33 m 31 m 32 m 33 ) ( j 11 j 21 j 31 m 11 m 21 m 31 ) ( j 12 j 22 j 32 m 12 m 22 m 32 ) ( j 13 j 23 j 33 m 13 m 23 m 33 ) ,
34.6.2 { j 11 j 12 j 13 j 21 j 22 j 23 j 31 j 32 j 33 } = j ( 1 ) 2 j ( 2 j + 1 ) { j 11 j 21 j 31 j 32 j 33 j } { j 12 j 22 j 32 j 21 j j 23 } { j 13 j 23 j 33 j j 11 j 12 } .
3: 34.7 Basic Properties: 9 j Symbol
34.7.1 { j 11 j 12 j 13 j 21 j 22 j 13 j 31 j 31 0 } = ( 1 ) j 12 + j 21 + j 13 + j 31 ( ( 2 j 13 + 1 ) ( 2 j 31 + 1 ) ) 1 2 { j 11 j 12 j 13 j 22 j 21 j 31 } .
34.7.2 j 12 j 34 ( 2 j 12 + 1 ) ( 2 j 34 + 1 ) ( 2 j 13 + 1 ) ( 2 j 24 + 1 ) { j 1 j 2 j 12 j 3 j 4 j 34 j 13 j 24 j } { j 1 j 2 j 12 j 3 j 4 j 34 j 13 j 24 j } = δ j 13 , j 13 δ j 24 , j 24 .
34.7.3 j 13 j 24 ( 1 ) 2 j 2 + j 24 + j 23 j 34 ( 2 j 13 + 1 ) ( 2 j 24 + 1 ) { j 1 j 2 j 12 j 3 j 4 j 34 j 13 j 24 j } { j 1 j 3 j 13 j 4 j 2 j 24 j 14 j 23 j } = { j 1 j 2 j 12 j 4 j 3 j 34 j 14 j 23 j } .
34.7.4 ( j 13 j 23 j 33 m 13 m 23 m 33 ) { j 11 j 12 j 13 j 21 j 22 j 23 j 31 j 32 j 33 } = m r 1 , m r 2 , r = 1 , 2 , 3 ( j 11 j 12 j 13 m 11 m 12 m 13 ) ( j 21 j 22 j 23 m 21 m 22 m 23 ) ( j 31 j 32 j 33 m 31 m 32 m 33 ) ( j 11 j 21 j 31 m 11 m 21 m 31 ) ( j 12 j 22 j 32 m 12 m 22 m 32 ) .
34.7.5 j ( 2 j + 1 ) { j 11 j 12 j j 21 j 22 j 23 j 31 j 32 j 33 } { j 11 j 12 j j 23 j 33 j } = ( 1 ) 2 j { j 21 j 22 j 23 j 12 j j 32 } { j 31 j 32 j 33 j j 11 j 21 } .
4: 32.3 Graphics
Plots of solutions w k ( x ) of P I  with w k ( 0 ) = 0 and w k ( 0 ) = k for various values of k , and the parabola 6 w 2 + x = 0 . … with β = 0 , α = 2 ν + 1 , and …If we set d 2 u / d x 2 = 0 in (32.3.2) and solve for u , then …
See accompanying text
Figure 32.3.7: u k ( x ; 1 2 ) for 12 x 4 with k = 0.33554 691 , 0.33554 692 . … Magnify
See accompanying text
Figure 32.3.8: u k ( x ; 1 2 ) for 12 x 4 with k = 0.47442 , 0.47443 . … Magnify
5: 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
π / 12 1 4 2 ( 3 1 ) 1 4 2 ( 3 + 1 ) 2 3 2 ( 3 + 1 ) 2 ( 3 1 ) 2 + 3
5 π / 12 1 4 2 ( 3 + 1 ) 1 4 2 ( 3 1 ) 2 + 3 2 ( 3 1 ) 2 ( 3 + 1 ) 2 3
7 π / 12 1 4 2 ( 3 + 1 ) 1 4 2 ( 3 1 ) ( 2 + 3 ) 2 ( 3 1 ) 2 ( 3 + 1 ) ( 2 3 )
4.17.3 lim z 0 1 cos z z 2 = 1 2 .
6: 10.48 Graphs
See accompanying text
Figure 10.48.1: 𝗃 n ( x ) , n = 0 ( 1 ) 4 , 0 x 12 . Magnify
See accompanying text
Figure 10.48.2: 𝗒 n ( x ) , n = 0 ( 1 ) 4 , 0 < x 12 . Magnify
See accompanying text
Figure 10.48.3: 𝗃 5 ( x ) , 𝗒 5 ( x ) , 𝗃 5 2 ( x ) + 𝗒 5 2 ( x ) , 0 x 12 . Magnify
See accompanying text
Figure 10.48.4: 𝗃 5 ( x ) , 𝗒 5 ( x ) , 𝗃 5 2 ( x ) + 𝗒 5 2 ( x ) , 0 x 12 . Magnify
See accompanying text
Figure 10.48.5: 𝗂 0 ( 1 ) ( x ) , 𝗂 0 ( 2 ) ( x ) , 𝗄 0 ( x ) , 0 x 4 . Magnify
7: 27.2 Functions
where p 1 , p 2 , , p ν ( n ) are the distinct prime factors of n , each exponent a r is positive, and ν ( n ) is the number of distinct primes dividing n . … The ϕ ( n ) numbers a , a 2 , , a ϕ ( n ) are relatively prime to n and distinct (mod n ). …It is the special case k = 2 of the function d k ( n ) that counts the number of ways of expressing n as the product of k factors, with the order of factors taken into account. …
27.2.12 μ ( n ) = { 1 , n = 1 , ( 1 ) ν ( n ) , a 1 = a 2 = = a ν ( n ) = 1 , 0 , otherwise .
Table 27.2.2: Functions related to division.
n ϕ ( n ) d ( n ) σ ( n ) n ϕ ( n ) d ( n ) σ ( n ) n ϕ ( n ) d ( n ) σ ( n ) n ϕ ( n ) d ( n ) σ ( n )
11 10 2 12 24 8 8 60 37 36 2 38 50 20 6 93
8: 34.11 Higher-Order 3 n j Symbols
For information on 12 j , 15 j ,…, symbols, see Varshalovich et al. (1988, §10.12) and Yutsis et al. (1962, pp. 62–65 and 122–153).
9: 23.17 Elementary Properties
λ ( i ) = 1 2 ,
23.17.5 1728 J ( τ ) = q 2 + 744 + 1 96884 q 2 + 214 93760 q 4 + ,
23.17.6 η ( τ ) = n = ( 1 ) n q ( 6 n + 1 ) 2 / 12 .
23.17.8 η ( τ ) = q 1 / 12 n = 1 ( 1 q 2 n ) ,
with q 1 / 12 = e i π τ / 12 .
10: 24.2 Definitions and Generating Functions
B 2 n + 1 = 0 ,
( 1 ) n + 1 B 2 n > 0 , n = 1 , 2 , .
E 2 n + 1 = 0 ,
( 1 ) n E 2 n > 0 .
24.2.9 E n = 2 n E n ( 1 2 ) = integer ,