About the Project

DΕΎrbasjan sum

AdvancedHelp

The term"rbasjan" was not found.Possible alternative term: "dΕΎrbasjan".

(0.002 seconds)

1—10 of 411 matching pages

1: 26.10 Integer Partitions: Other Restrictions
β–Ί p ⁑ ( π’Ÿ , n ) denotes the number of partitions of n into distinct parts. p m ⁑ ( π’Ÿ , n ) denotes the number of partitions of n into at most m distinct parts. p ⁑ ( π’Ÿ ⁒ k , n ) denotes the number of partitions of n into parts with difference at least k . …If more than one restriction applies, then the restrictions are separated by commas, for example, p ⁑ ( π’Ÿ ⁒ 2 , T , n ) . … β–ΊNote that p ⁑ ( π’Ÿ ⁒ 3 , n ) p ⁑ ( π’Ÿ ⁒ 3 , n ) , with strict inequality for n 9 . …
2: 28.25 Asymptotic Expansions for Large ⁑ z
β–Ί
28.25.1 M Ξ½ ( 3 , 4 ) ⁑ ( z , h ) e ± i ⁒ ( 2 ⁒ h ⁒ cosh ⁑ z ( 1 2 ⁒ Ξ½ + 1 4 ) ⁒ Ο€ ) ( Ο€ ⁒ h ⁒ ( cosh ⁑ z + 1 ) ) 1 2 ⁒ m = 0 D m ± ( βˆ“ 4 ⁒ i ⁒ h ⁒ ( cosh ⁑ z + 1 ) ) m ,
β–Ί
D 1 ± = 0 ,
β–Ί
D 0 ± = 1 ,
β–Ί
28.25.3 ( m + 1 ) ⁒ D m + 1 ± + ( ( m + 1 2 ) 2 ± ( m + 1 4 ) ⁒ 8 ⁒ i ⁒ h + 2 ⁒ h 2 a ) ⁒ D m ± ± ( m 1 2 ) ⁒ ( 8 ⁒ i ⁒ h ⁒ m ) ⁒ D m 1 ± = 0 , m 0 .
3: 1.10 Functions of a Complex Variable
β–ΊLet D be a bounded domain with boundary D and let D ¯ = D D . … β–ΊIf 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 . … β–ΊLet F ⁑ ( z ) be a multivalued function and D be a domain. … β–ΊSuppose D is a domain, and …
4: 19.21 Connection Formulas
β–ΊThe complete case of R J can be expressed in terms of R F and R D : … β–Ί 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 … β–Ί
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 ,
β–ΊBecause 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)). β–Ί
19.21.11 6 ⁒ R G ⁑ ( x , y , z ) = 3 ⁒ ( x + y + z ) ⁒ R F ⁑ ( x , y , z ) x 2 ⁒ R D ⁑ ( y , z , x ) = x ⁒ ( y + z ) ⁒ R D ⁑ ( y , z , x ) ,
5: 26.6 Other Lattice Path Numbers
β–Ί
Delannoy Number D ⁑ ( m , n )
β–Ί 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 ) . β–Ί
26.6.1 D ⁑ ( m , n ) = k = 0 n ( n k ) ⁒ ( m + n k n ) = k = 0 n 2 k ⁒ ( m k ) ⁒ ( n k ) .
β–Ί
26.6.5 m , n = 0 D ⁑ ( m , n ) ⁒ x m ⁒ y n = 1 1 x y x ⁒ y ,
β–Ί
26.6.6 n = 0 D ⁑ ( n , n ) ⁒ x n = 1 1 6 ⁒ x + x 2 ,
6: 1.16 Distributions
β–ΊWe denote it by π’Ÿ ⁑ ( I ) . … β–Ίfor all Ο• π’Ÿ ⁑ ( I ) . … β–ΊIf f is a locally integrable function then its distributional derivative is 𝐷 f = Ξ› f . … β–ΊThe distributional derivative 𝐷 k f of f is defined by …
7: 29.6 Fourier Series
β–Ί
29.6.38 𝐸𝑠 Ξ½ 2 ⁒ m + 1 ⁑ ( z , k 2 ) = dn ⁑ ( z , k ) ⁒ p = 0 D 2 ⁒ p + 1 ⁒ sin ⁑ ( ( 2 ⁒ p + 1 ) ⁒ Ο• ) ,
β–Ί
29.6.42 ( 1 1 2 ⁒ k 2 ) ⁒ p = 0 D 2 ⁒ p + 1 2 + 1 2 ⁒ k 2 ⁒ ( 1 2 ⁒ D 1 2 p = 0 D 2 ⁒ p + 1 ⁒ D 2 ⁒ p + 3 ) = 1 ,
β–Ί
29.6.43 p = 0 ( 2 ⁒ p + 1 ) ⁒ D 2 ⁒ p + 1 > 0 ,
β–Ί
29.6.57 ( 1 1 2 ⁒ k 2 ) ⁒ p = 1 D 2 ⁒ p 2 1 2 ⁒ k 2 ⁒ p = 1 D 2 ⁒ p ⁒ D 2 ⁒ p + 2 = 1 ,
β–Ί
29.6.58 p = 0 ( 2 ⁒ p + 2 ) ⁒ D 2 ⁒ p + 2 > 0 ,
8: 1.11 Zeros of Polynomials
β–Ί
D = b 2 4 ⁒ a ⁒ c .
β–ΊThe sum and product of the roots are respectively b / a and c / a . … β–ΊLet β–Ί
D 1 = a 1 ,
β–ΊThen 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 .
9: 29.15 Fourier Series and Chebyshev Series
β–Ί
29.15.34 [ D 1 , D 3 , , D 2 ⁒ n + 1 ] T ,
β–Ί
29.15.35 ( 1 1 2 ⁒ k 2 ) ⁒ p = 0 n D 2 ⁒ p + 1 2 + 1 2 ⁒ k 2 ⁒ ( 1 2 ⁒ D 1 2 p = 0 n 1 D 2 ⁒ p + 1 ⁒ D 2 ⁒ p + 3 ) = 1 ,
β–Ί
29.15.36 p = 0 n ( 2 ⁒ p + 1 ) ⁒ D 2 ⁒ p + 1 > 0 .
β–Ί
29.15.40 ( 1 1 2 ⁒ k 2 ) ⁒ p = 0 n D 2 ⁒ p + 2 2 1 2 ⁒ k 2 ⁒ p = 1 n D 2 ⁒ p ⁒ D 2 ⁒ p + 2 = 1 ,
β–Ί
29.15.41 p = 0 n ( 2 ⁒ p + 2 ) ⁒ D 2 ⁒ p + 2 > 0 .
10: 10.20 Uniform Asymptotic Expansions for Large Order
β–ΊIn the following formulas for the coefficients A k ⁑ ( ΞΆ ) , B k ⁑ ( ΞΆ ) , C k ⁑ ( ΞΆ ) , and D k ⁑ ( ΞΆ ) , u k , v k are the constants defined in §9.7(i), and U k ⁑ ( p ) , V k ⁑ ( p ) are the polynomials in p of degree 3 ⁒ k defined in §10.41(ii). … β–ΊNote: Another way of arranging the above formulas for the coefficients A k ⁑ ( ΞΆ ) , B k ⁑ ( ΞΆ ) , C k ⁑ ( ΞΆ ) , and D k ⁑ ( ΞΆ ) would be by analogy with (12.10.42) and (12.10.46). … β–ΊEach of the coefficients A k ⁑ ( ΞΆ ) , B k ⁑ ( ΞΆ ) , C k ⁑ ( ΞΆ ) , and D k ⁑ ( ΞΆ ) , k = 0 , 1 , 2 , , is real and infinitely differentiable on the interval < ΞΆ < . … β–ΊFor numerical tables of ΞΆ = ΞΆ ⁑ ( z ) , ( 4 ⁒ ΞΆ / ( 1 z 2 ) ) 1 4 and A k ⁑ ( ΞΆ ) , B k ⁑ ( ΞΆ ) , C k ⁑ ( ΞΆ ) , and D k ⁑ ( ΞΆ ) see Olver (1962, pp. 28–42). … β–ΊThe equations of the curved boundaries D 1 ⁒ E 1 and D 2 ⁒ E 2 in the ΞΆ -plane are given parametrically by …