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1: 34.2 Definition: 3 j Symbol
§34.2 Definition: 3 j Symbol
The quantities j 1 , j 2 , j 3 in the 3 j symbol are called angular momenta. …They therefore satisfy the triangle conditions …where r , s , t is any permutation of 1 , 2 , 3 . The corresponding projective quantum numbers m 1 , m 2 , m 3 are given by …
2: 26.10 Integer Partitions: Other Restrictions
p ( 𝒟 , n ) denotes the number of partitions of n into distinct parts. p m ( 𝒟 , n ) denotes the number of partitions of n into at most m distinct parts. … p ( 𝒟 3 , n ) denotes the number of partitions of n into parts with difference at least 3, except that multiples of 3 must differ by at least 6. … Note that p ( 𝒟 3 , n ) p ( 𝒟 3 , n ) , with strict inequality for n 9 . It is known that for k > 3 , p ( 𝒟 k , n ) p ( A 1 , k + 3 , n ) , with strict inequality for n sufficiently large, provided that k = 2 m 1 , m = 3 , 4 , 5 , or k 32 ; see Yee (2004). …
3: 26.6 Other Lattice Path Numbers
Delannoy Number D ( m , n )
D ( m , n ) is the number of paths from ( 0 , 0 ) to ( m , n ) that are composed of directed line segments of the form ( 1 , 0 ) , ( 0 , 1 ) , or ( 1 , 1 ) . …
Table 26.6.1: Delannoy numbers D ( m , n ) .
m n
26.6.4 r ( n ) = D ( n , n ) D ( n + 1 , n 1 ) , n 1 .
26.6.10 D ( m , n ) = D ( m , n 1 ) + D ( m 1 , n ) + D ( m 1 , n 1 ) , m , n 1 ,
4: 28.25 Asymptotic Expansions for Large z
28.25.1 M ν ( 3 , 4 ) ( z , h ) e ± i ( 2 h cosh z ( 1 2 ν + 1 4 ) π ) ( π h ( cosh z + 1 ) ) 1 2 m = 0 D m ± ( 4 i h ( cosh z + 1 ) ) m ,
D 1 ± = 0 ,
D 0 ± = 1 ,
The upper signs correspond to M ν ( 3 ) ( z , h ) and the lower signs to M ν ( 4 ) ( z , h ) . The expansion (28.25.1) is valid for M ν ( 3 ) ( z , h ) when …
5: 1.11 Zeros of Polynomials
Set z = w 1 3 a to reduce f ( z ) = z 3 + a z 2 + b z + c to g ( w ) = w 3 + p w + q , with p = ( 3 b a 2 ) / 3 , q = ( 2 a 3 9 a b + 27 c ) / 27 . … f ( z ) = z 3 6 z 2 + 6 z 2 , g ( w ) = w 3 6 w 6 , A = 3 4 3 , B = 3 2 3 . Roots of f ( z ) = 0 are 2 + 4 3 + 2 3 , 2 + 4 3 ρ + 2 3 ρ 2 , 2 + 4 3 ρ 2 + 2 3 ρ . … Let … Then f ( z ) , with a n 0 , is stable iff a 0 0 ; D 2 k > 0 , k = 1 , , 1 2 n ; sign D 2 k + 1 = sign a 0 , k = 0 , 1 , , 1 2 n 1 2 .
6: 28.8 Asymptotic Expansions for Large q
28.8.1 a m ( h 2 ) b m + 1 ( h 2 ) } 2 h 2 + 2 s h 1 8 ( s 2 + 1 ) 1 2 7 h ( s 3 + 3 s ) 1 2 12 h 2 ( 5 s 4 + 34 s 2 + 9 ) 1 2 17 h 3 ( 33 s 5 + 410 s 3 + 405 s ) 1 2 20 h 4 ( 63 s 6 + 1260 s 4 + 2943 s 2 + 486 ) 1 2 25 h 5 ( 527 s 7 + 15617 s 5 + 69001 s 3 + 41607 s ) + .
Also let ξ = 2 h cos x and D m ( ξ ) = e ξ 2 / 4 𝐻𝑒 m ( ξ ) 18.3). …
28.8.4 U m ( ξ ) D m ( ξ ) 1 2 6 h ( D m + 4 ( ξ ) 4 ! ( m 4 ) D m 4 ( ξ ) ) + 1 2 13 h 2 ( D m + 8 ( ξ ) 2 5 ( m + 2 ) D m + 4 ( ξ ) + 4 !  2 5 ( m 1 ) ( m 4 ) D m 4 ( ξ ) + 8 ! ( m 8 ) D m 8 ( ξ ) ) + ,
28.8.5 V m ( ξ ) 1 2 4 h ( D m + 2 ( ξ ) m ( m 1 ) D m 2 ( ξ ) ) + 1 2 10 h 2 ( D m + 6 ( ξ ) + ( m 2 25 m 36 ) D m + 2 ( ξ ) m ( m 1 ) ( m 2 + 27 m 10 ) D m 2 ( ξ ) 6 ! ( m 6 ) D m 6 ( ξ ) ) + ,
σ m 1 + s 2 3 h + 4 s 2 + 3 2 7 h 2 + 19 s 3 + 59 s 2 11 h 3 + ,
7: 4.43 Cubic Equations
A = ( 4 3 p ) 1 / 2 ,
4.43.2 z 3 + p z + q = 0
  • (a)

    A sin a , A sin ( a + 2 3 π ) , and A sin ( a + 4 3 π ) , with sin ( 3 a ) = 4 q / A 3 , when 4 p 3 + 27 q 2 0 .

  • (b)

    A cosh a , A cosh ( a + 2 3 π i ) , and A cosh ( a + 4 3 π i ) , with cosh ( 3 a ) = 4 q / A 3 , when p < 0 , q < 0 , and 4 p 3 + 27 q 2 > 0 .

  • (c)

    B sinh a , B sinh ( a + 2 3 π i ) , and B sinh ( a + 4 3 π i ) , with sinh ( 3 a ) = 4 q / B 3 , when p > 0 .

  • 8: 19.21 Connection Formulas
    The complete case of R J can be expressed in terms of R F and R D : … R D ( x , y , z ) is symmetric only in x and y , but either (nonzero) x or (nonzero) y can be moved to the third position by using …
    19.21.8 R D ( y , z , x ) + R D ( z , x , y ) + R D ( x , y , z ) = 3 x 1 / 2 y 1 / 2 z 1 / 2 ,
    19.21.9 x R D ( y , z , x ) + y R D ( z , x , y ) + z R D ( x , y , z ) = 3 R F ( x , y , z ) .
    Because R G is completely symmetric, x , y , z can be permuted on the right-hand side of (19.21.10) so that ( x z ) ( y z ) 0 if the variables are real, thereby avoiding cancellations when R G is calculated from R F and R D (see §19.36(i)). …
    9: 1.10 Functions of a Complex Variable
    Let D be a bounded domain with boundary D and let D ¯ = D D . … If u ( z ) is harmonic in D , z 0 D , and u ( z ) u ( z 0 ) for all z D , then u ( z ) is constant in D . Moreover, if D is bounded and u ( z ) is continuous on D ¯ and harmonic in D , then u ( z ) is maximum at some point on D . … Let F ( z ) be a multivalued function and D be a domain. … Suppose D is a domain, and …
    10: 12.19 Tables
  • Kireyeva and Karpov (1961) includes D p ( x ( 1 + i ) ) for ± x = 0 ( .1 ) 5 , p = 0 ( .1 ) 2 , and ± x = 5 ( .01 ) 10 , p = 0 ( .5 ) 2 , 7D.

  • Karpov and Čistova (1964) includes D p ( x ) for p = 2 ( .1 ) 0 , ± x = 0 ( .01 ) 5 ; p = 2 ( .05 ) 0 , ± x = 5 ( .01 ) 10 , 6D.

  • Karpov and Čistova (1968) includes e 1 4 x 2 D p ( x ) and e 1 4 x 2 D p ( i x ) for x = 0 ( .01 ) 5 and x 1 = 0(.001 or .0001)5, p = 1 ( .1 ) 1 , 7D or 8S.

  • Murzewski and Sowa (1972) includes D n ( x ) ( = U ( n 1 2 , x ) ) for n = 1 ( 1 ) 20 , x = 0 ( .05 ) 3 , 7S.