About the Project

by parts

AdvancedHelp

(0.002 seconds)

1—10 of 297 matching pages

1: 26.11 Integer Partitions: Compositions
β–Ί 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: … β–Ί
26.11.2 c m ⁑ ( 0 ) = δ 0 , m ,
β–Ί
26.11.3 c m ⁑ ( n ) = ( n 1 m 1 ) ,
2: 27.4 Euler Products and Dirichlet Series
β–Ί
27.4.3 ΢ ⁑ ( s ) = n = 1 n s = p ( 1 p s ) 1 , ⁑ s > 1 .
β–Ί
27.4.5 n = 1 μ ⁑ ( n ) ⁒ n s = 1 ΢ ⁑ ( s ) , ⁑ s > 1 ,
β–Ί
27.4.6 n = 1 Ο• ⁑ ( n ) ⁒ n s = ΞΆ ⁑ ( s 1 ) ΞΆ ⁑ ( s ) , ⁑ s > 2 ,
β–Ί
27.4.10 n = 1 d k ⁑ ( n ) ⁒ n s = ( ΢ ⁑ ( s ) ) k , ⁑ s > 1 ,
β–Ί
27.4.11 n = 1 Οƒ Ξ± ⁑ ( n ) ⁒ n s = ΞΆ ⁑ ( s ) ⁒ ΞΆ ⁑ ( s Ξ± ) , ⁑ s > max ⁑ ( 1 , 1 + ⁑ Ξ± ) ,
3: 7.9 Continued Fractions
β–Ί
7.9.1 Ο€ ⁒ e z 2 ⁒ erfc ⁑ z = z z 2 + 1 2 1 + 1 z 2 + 3 2 1 + 2 z 2 + β‹― , ⁑ z > 0 ,
β–Ί
7.9.2 Ο€ ⁒ e z 2 ⁒ erfc ⁑ z = 2 ⁒ z 2 ⁒ z 2 + 1 1 2 2 ⁒ z 2 + 5 3 4 2 ⁒ z 2 + 9 β‹― , ⁑ z > 0 ,
β–Ί
7.9.3 w ⁑ ( z ) = i Ο€ ⁒ 1 z 1 2 z 1 z 3 2 z 2 z β‹― , ⁑ z > 0 .
4: 8.14 Integrals
β–Ί
8.14.1 0 e a ⁒ x ⁒ Ξ³ ⁑ ( b , x ) Ξ“ ⁑ ( b ) ⁒ d x = ( 1 + a ) b a , ⁑ a > 0 , ⁑ b > 1 ,
β–Ί
8.14.2 0 e a ⁒ x ⁒ Ξ“ ⁑ ( b , x ) ⁒ d x = Ξ“ ⁑ ( b ) ⁒ 1 ( 1 + a ) b a , ⁑ a > 1 , ⁑ b > 1 .
β–Ί
8.14.3 0 x a 1 ⁒ Ξ³ ⁑ ( b , x ) ⁒ d x = Ξ“ ⁑ ( a + b ) a , ⁑ a < 0 , ⁑ ( a + b ) > 0 ,
β–Ί
8.14.4 0 x a 1 ⁒ Ξ“ ⁑ ( b , x ) ⁒ d x = Ξ“ ⁑ ( a + b ) a , ⁑ a > 0 , ⁑ ( a + b ) > 0 ,
β–Ί
8.14.6 0 x a 1 ⁒ e s ⁒ x ⁒ Ξ“ ⁑ ( b , x ) ⁒ d x = Ξ“ ⁑ ( 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
β–ΊPart 2. …
6: 26.21 Tables
β–ΊAbramowitz 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 ⁑ ( π’Ÿ , n ) for n up to 500. β–ΊAndrews (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
β–Ί
28.25.4 ⁑ z + , Ο€ + Ξ΄ ph ⁑ h + ⁑ z 2 ⁒ Ο€ Ξ΄ ,
β–Ί
28.25.5 ⁑ z + , 2 ⁒ Ο€ + Ξ΄ ph ⁑ h + ⁑ z Ο€ Ξ΄ ,
8: 19.3 Graphics
β–Ί
β–Ί
See accompanying text
β–Ί
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
β–Ί
β–Ί
See accompanying text
β–Ί
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
β–Ί
β–Ί
See accompanying text
β–Ί
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
β–Ί
β–Ί
See accompanying text
β–Ί
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
β–Ί
13.10.3 0 e z ⁒ t ⁒ t b 1 ⁒ 𝐌 ⁑ ( a , c , k ⁒ t ) ⁒ d t = Ξ“ ⁑ ( b ) ⁒ z b ⁒ 𝐅 1 2 ⁑ ( a , b ; c ; k / z ) , ⁑ b > 0 , ⁑ z > max ⁑ ( ⁑ k , 0 ) ,
β–Ί
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 ,
β–Ί
13.10.5 0 e t ⁒ t b 1 ⁒ 𝐌 ⁑ ( a , c , t ) ⁒ d t = Ξ“ ⁑ ( b ) ⁒ Ξ“ ⁑ ( c a b ) Ξ“ ⁑ ( c a ) ⁒ Ξ“ ⁑ ( c b ) , ⁑ ( c a ) > ⁑ b > 0 ,
β–Ί
13.10.7 0 e z ⁒ t ⁒ t b 1 ⁒ U ⁑ ( a , c , t ) ⁒ d t = Ξ“ ⁑ ( b ) ⁒ Ξ“ ⁑ ( b c + 1 ) ⁒ z b ⁒ 𝐅 1 2 ⁑ ( a , b ; a + b c + 1 ; 1 1 z ) , ⁑ b > max ⁑ ( ⁑ c 1 , 0 ) , ⁑ z > 0 .
β–Ί
13.10.10 0 t Ξ» 1 ⁒ 𝐌 ⁑ ( a , b , t ) ⁒ d t = Ξ“ ⁑ ( Ξ» ) ⁒ Ξ“ ⁑ ( a Ξ» ) Ξ“ ⁑ ( a ) ⁒ Ξ“ ⁑ ( b Ξ» ) , 0 < ⁑ Ξ» < ⁑ a ,
10: 7.14 Integrals
β–Ί
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 ⁒ Ο€ ,
β–Ί
7.14.3 0 e a ⁒ t ⁒ erf ⁑ b ⁒ t ⁒ d t = 1 a ⁒ b a + b , ⁑ a > 0 , ⁑ b > 0 ,
β–Ί
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 ⁒ Ο€ , ⁑ b > 0 , ⁑ c 0 .
β–Ί
7.14.5 0 e a ⁒ t ⁒ C ⁑ ( t ) ⁒ d t = 1 a ⁒ f ⁑ ( a Ο€ ) , ⁑ a > 0 ,
β–Ί
7.14.6 0 e a ⁒ t ⁒ S ⁑ ( t ) ⁒ d t = 1 a ⁒ g ⁑ ( a Ο€ ) , ⁑ a > 0 ,