About the Project

infinite series

AdvancedHelp

(0.003 seconds)

11—20 of 67 matching pages

11: 25.13 Periodic Zeta Function
25.13.1 F ( x , s ) n = 1 e 2 π i n x n s ,
12: Bibliography V
  • R. S. Varma (1941) An infinite series of Weber’s parabolic cylinder functions. Proc. Benares Math. Soc. (N.S.) 3, pp. 37.
  • A. N. Vavreck and W. Thompson (1984) Some novel infinite series of spherical Bessel functions. Quart. Appl. Math. 42 (3), pp. 321–324.
  • 13: 25.6 Integer Arguments
    25.6.8 ζ ( 2 ) = 3 k = 1 1 k 2 ( 2 k k ) .
    25.6.9 ζ ( 3 ) = 5 2 k = 1 ( 1 ) k 1 k 3 ( 2 k k ) .
    25.6.10 ζ ( 4 ) = 36 17 k = 1 1 k 4 ( 2 k k ) .
    25.6.12 ζ ′′ ( 0 ) = 1 2 ( ln ( 2 π ) ) 2 + 1 2 γ 2 1 24 π 2 + γ 1 ,
    14: 5.19 Mathematical Applications
    As shown in Temme (1996b, §3.4), the results given in §5.7(ii) can be used to sum infinite series of rational functions. …
    15: 31.11 Expansions in Series of Hypergeometric Functions
    §31.11(v) Doubly-Infinite Series
    Schmidt (1979) gives expansions of path-multiplicative solutions (§31.6) in terms of doubly-infinite series of hypergeometric functions. …
    16: 15.19 Methods of Computation
    However, by appropriate choice of the constant z 0 in (15.15.1) we can obtain an infinite series that converges on a disk containing z = e ± π i / 3 . …
    17: 25.11 Hurwitz Zeta Function
    25.11.8 ζ ( s , 1 2 a ) = ζ ( s , 1 2 a + 1 2 ) + 2 s n = 0 ( 1 ) n ( n + a ) s , s > 0 , s 1 , 0 < a 1 .
    25.11.9 ζ ( 1 s , a ) = 2 Γ ( s ) ( 2 π ) s n = 1 1 n s cos ( 1 2 π s 2 n π a ) , s > 0 if 0 < a < 1 ; s > 1 if a = 1 .
    25.11.35 n = 0 ( 1 ) n ( n + a ) s = 1 Γ ( s ) 0 x s 1 e a x 1 + e x d x = 2 s ( ζ ( s , 1 2 a ) ζ ( s , 1 2 ( 1 + a ) ) ) , a > 0 , s > 0 ; or a = 0 , a 0 , 0 < s < 1 .
    25.11.38 k = 1 ( n + k k ) ζ ( n + k + 1 , a ) z k = ( 1 ) n n ! ( ψ ( n ) ( a ) ψ ( n ) ( a z ) ) , n = 1 , 2 , 3 , , a > 0 , | z | < | a | .
    25.11.39 k = 2 k 2 k ζ ( k + 1 , 3 4 ) = 8 G ,
    18: 25.15 Dirichlet L -functions
    25.15.1 L ( s , χ ) = n = 1 χ ( n ) n s , s > 1 ,
    19: 1.15 Summability Methods
    §1.15 Summability Methods
    20: 25.14 Lerch’s Transcendent
    25.14.1 Φ ( z , s , a ) n = 0 z n ( a + n ) s , | z | < 1 ; s > 1 , | z | = 1 .