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1: 26.11 Integer Partitions: Compositions
c ( n ) denotes the number of compositions of n , and c m ( n ) is the number of compositions into exactly m parts. c ( T , n ) is the number of compositions of n with no 1’s, where again T = { 2 , 3 , 4 , } . The integer 0 is considered to have one composition consisting of no parts: …
26.11.2 c m ( 0 ) = δ 0 , m ,
2: 27.4 Euler Products and Dirichlet Series
27.4.3 ζ ( s ) = n = 1 n - s = p ( 1 - p - s ) - 1 , s > 1 .
27.4.5 n = 1 μ ( n ) n - s = 1 ζ ( s ) , s > 1 ,
27.4.6 n = 1 ϕ ( n ) n - s = ζ ( s - 1 ) ζ ( s ) , s > 2 ,
27.4.7 n = 1 λ ( n ) n - s = ζ ( 2 s ) ζ ( s ) , s > 1 ,
27.4.11 n = 1 σ α ( n ) n - s = ζ ( s ) ζ ( s - α ) , s > max ( 1 , 1 + α ) ,
3: 7.9 Continued Fractions
7.9.1 π e z 2 erfc z = z z 2 + 1 2 1 + 1 z 2 + 3 2 1 + 2 z 2 + , z > 0 ,
7.9.2 π e z 2 erfc z = 2 z 2 z 2 + 1 - 1 2 2 z 2 + 5 - 3 4 2 z 2 + 9 - , z > 0 ,
7.9.3 w ( z ) = i π 1 z - 1 2 z - 1 z - 3 2 z - 2 z - , z > 0 .
4: 8.14 Integrals
8.14.1 0 e - a x γ ( b , x ) Γ ( b ) d x = ( 1 + a ) - b a , a > 0 , b > - 1 ,
8.14.2 0 e - a x Γ ( b , x ) d x = Γ ( b ) 1 - ( 1 + a ) - b a , a > - 1 , b > - 1 .
8.14.3 0 x a - 1 γ ( b , x ) d x = - Γ ( a + b ) a , a < 0 , ( a + b ) > 0 ,
8.14.4 0 x a - 1 Γ ( b , x ) d x = Γ ( a + b ) a , a > 0 , ( a + b ) > 0 ,
8.14.6 0 x a - 1 e - s x Γ ( b , x ) d x = Γ ( a + b ) a ( 1 + s ) a + b F ( 1 , a + b ; 1 + a ; s / ( 1 + s ) ) , s > - 1 , ( a + b ) > 0 , a > 0 .
5: Sidebar 21.SB1: Periodic Surface Waves
Part 2. …
6: 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. Andrews (1976) contains tables of the number of unrestricted partitions, partitions into odd parts, partitions into parts ± 2 ( mod 5 ) , partitions into parts ± 1 ( mod 5 ) , and unrestricted plane partitions up to 100. …
7: 28.25 Asymptotic Expansions for Large z
§28.25 Asymptotic Expansions for Large z
28.25.4 z + , - π + δ ph h + z 2 π - δ ,
28.25.5 z + , - 2 π + δ ph h + z π - δ ,
8: 19.3 Graphics
See accompanying text
Figure 19.3.9: ( K ( k ) ) as a function of complex k 2 for - 2 ( k 2 ) 2 , - 2 ( k 2 ) 2 . The real part is symmetric under reflection in the real axis. … Magnify 3D Help
See accompanying text
Figure 19.3.10: ( K ( k ) ) as a function of complex k 2 for - 2 ( k 2 ) 2 , - 2 ( k 2 ) 2 . The imaginary part is 0 for k 2 < 1 , and is antisymmetric under reflection in the real axis. … Magnify 3D Help
See accompanying text
Figure 19.3.11: ( E ( k ) ) as a function of complex k 2 for - 2 ( k 2 ) 2 , - 2 ( k 2 ) 2 . The real part is symmetric under reflection in the real axis. … Magnify 3D Help
See accompanying text
Figure 19.3.12: ( E ( k ) ) as a function of complex k 2 for - 2 ( k 2 ) 2 , - 2 ( k 2 ) 2 . The imaginary part is 0 for k 2 1 and is antisymmetric under reflection in the real axis. … Magnify 3D Help
9: 13.10 Integrals
13.10.3 0 e - z t t b - 1 M ( a , c , k t ) d t = Γ ( b ) z - b F 1 2 ( a , b ; c ; k / z ) , b > 0 , z > max ( k , 0 ) ,
13.10.4 0 e - z t t b - 1 M ( a , b , t ) d t = z - b ( 1 - 1 z ) - a , b > 0 , z > 1 ,
13.10.5 0 e - t t b - 1 M ( a , c , t ) d t = Γ ( b ) Γ ( c - a - b ) Γ ( c - a ) Γ ( c - b ) , ( c - a ) > b > 0 ,
13.10.10 0 t λ - 1 M ( a , b , - t ) d t = Γ ( λ ) Γ ( a - λ ) Γ ( a ) Γ ( b - λ ) , 0 < λ < a ,
13.10.11 0 t λ - 1 U ( a , b , t ) d t = Γ ( λ ) Γ ( a - λ ) Γ ( λ - b + 1 ) Γ ( a ) Γ ( a - b + 1 ) , max ( b - 1 , 0 ) < λ < a .
10: 7.14 Integrals
7.14.2 0 e - a t erf ( b t ) d t = 1 a e a 2 / ( 4 b 2 ) erfc ( a 2 b ) , a > 0 , | ph b | < 1 4 π ,
7.14.3 0 e - a t erf b t d t = 1 a b a + b , a > 0 , b > 0 ,
7.14.4 0 e ( a - b ) t erfc ( a t + c t ) d t = e - 2 ( a c + b c ) b ( a + b ) , | ph a | < 1 2 π , b > 0 , c 0 .
7.14.5 0 e - a t C ( t ) d t = 1 a f ( a π ) , a > 0 ,
7.14.6 0 e - a t S ( t ) d t = 1 a g ( a π ) , a > 0 ,