About the Project
NIST

summation by parts

AdvancedHelp

(0.001 seconds)

1—10 of 16 matching pages

1: 2.10 Sums and Sequences
§2.10(ii) Summation by Parts
The formula for summation by parts is …
2: 1.8 Fourier Series
1.8.16 n = - e - ( n + x ) 2 ω = π ω ( 1 + 2 n = 1 e - n 2 π 2 / ω cos ( 2 n π x ) ) , ω > 0 .
3: 25.11 Hurwitz Zeta Function
25.11.28 ζ ( s , a ) = 1 2 a - s + a 1 - s s - 1 + k = 1 n B 2 k ( 2 k ) ! ( s ) 2 k - 1 a 1 - s - 2 k + 1 Γ ( s ) 0 ( 1 e x - 1 - 1 x + 1 2 - k = 1 n B 2 k ( 2 k ) ! x 2 k - 1 ) x s - 1 e - a x d x , s > - ( 2 n + 1 ) , s 1 , a > 0 .
4: Bibliography B
  • Á. Baricz and T. K. Pogány (2013) Integral representations and summations of the modified Struve function. Acta Math. Hungar. 141 (3), pp. 254–281.
  • B. C. Berndt (1975a) Character analogues of the Poisson and Euler-MacLaurin summation formulas with applications. J. Number Theory 7 (4), pp. 413–445.
  • B. C. Berndt (1975b) Periodic Bernoulli numbers, summation formulas and applications. In Theory and Application of Special Functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975), pp. 143–189.
  • B. C. Berndt (1989) Ramanujan’s Notebooks. Part II. Springer-Verlag, New York.
  • J. G. Byatt-Smith (2000) The Borel transform and its use in the summation of asymptotic expansions. Stud. Appl. Math. 105 (2), pp. 83–113.
  • 5: 18.17 Integrals
    18.17.34 0 e - x z L n ( α ) ( x ) e - x x α d x = Γ ( α + n + 1 ) z n n ! ( z + 1 ) α + n + 1 , z > - 1 .
    18.17.38 0 1 P 2 n ( x ) x z - 1 d x = ( - 1 ) n ( 1 2 - 1 2 z ) n 2 ( 1 2 z ) n + 1 , z > 0 ,
    18.17.39 0 1 P 2 n + 1 ( x ) x z - 1 d x = ( - 1 ) n ( 1 - 1 2 z ) n 2 ( 1 2 + 1 2 z ) n + 1 , z > - 1 .
    18.17.40 0 e - a x L n ( α ) ( b x ) x z - 1 d x = Γ ( z + n ) n ! ( a - b ) n a - n - z F 1 2 ( - n , 1 + α - z 1 - n - z ; a a - b ) , a > 0 , z > 0 .
    18.17.41 0 e - a x He n ( x ) x z - 1 d x = Γ ( z + n ) a - n - 2 F 2 2 ( - 1 2 n , - 1 2 n + 1 2 - 1 2 z - 1 2 n , - 1 2 z - 1 2 n + 1 2 ; - 1 2 a 2 ) , a > 0 . Also, z > 0 , n even; z > - 1 , n odd.
    6: 25.5 Integral Representations
    25.5.7 ζ ( s ) = 1 2 + 1 s - 1 + m = 1 n B 2 m ( 2 m ) ! ( s ) 2 m - 1 + 1 Γ ( s ) 0 ( 1 e x - 1 - 1 x + 1 2 - m = 1 n B 2 m ( 2 m ) ! x 2 m - 1 ) x s - 1 e x d x , s > - ( 2 n + 1 ) , n = 1 , 2 , 3 , .
    7: Bibliography Z
  • Zeilberger (website) Doron Zeilberger’s Maple Packages and Programs Department of Mathematics, Rutgers University, New Jersey.
  • I. J. Zucker (1979) The summation of series of hyperbolic functions. SIAM J. Math. Anal. 10 (1), pp. 192–206.
  • M. I. Žurina and L. N. Karmazina (1964) Tables of the Legendre functions P - 1 / 2 + i τ ( x ) . Part I. Translated by D. E. Brown. Mathematical Tables Series, Vol. 22, Pergamon Press, Oxford.
  • M. I. Žurina and L. N. Karmazina (1965) Tables of the Legendre functions P - 1 / 2 + i τ ( x ) . Part II. Translated by Prasenjit Basu. Mathematical Tables Series, Vol. 38. A Pergamon Press Book, The Macmillan Co., New York.
  • 8: 35.4 Partitions and Zonal Polynomials
    Summation
    For T Ω and ( a ) , ( b ) > 1 2 ( m - 1 ) , …
    9: 8.12 Uniform Asymptotic Expansions for Large Parameter
    8.12.18 Q ( a , z ) P ( a , z ) } z a - 1 2 e - z Γ ( a ) ( d ( ± χ ) k = 0 A k ( χ ) z k / 2 k = 1 B k ( χ ) z k / 2 ) ,
    for z in | ph z | < 1 2 π , with ( z - a ) 0 for P ( a , z ) and ( z - a ) 0 for Q ( a , z ) . …
    10: Bibliography C
  • C. W. Clenshaw, D. W. Lozier, F. W. J. Olver, and P. R. Turner (1986) Generalized exponential and logarithmic functions. Comput. Math. Appl. Part B 12 (5-6), pp. 1091–1101.
  • C. W. Clenshaw (1955) A note on the summation of Chebyshev series. Math. Tables Aids Comput. 9 (51), pp. 118–120.