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11: Software Index
12: 9.10 Integrals
§9.10(viii) Repeated Integrals
13: 25.2 Definition and Expansions
25.2.9 ζ ( s ) = k = 1 N 1 k s + N 1 - s s - 1 - 1 2 N - s + k = 1 n ( s + 2 k - 2 2 k - 1 ) B 2 k 2 k N 1 - s - 2 k - ( s + 2 n 2 n + 1 ) N B ~ 2 n + 1 ( x ) x s + 2 n + 1 d x , s > - 2 n ; n , N = 1 , 2 , 3 , .
14: 2.6 Distributional Methods
To assign a distribution to the function f n ( t ) , we first let f n , n ( t ) denote the n th repeated integral (§1.4(v)) of f n :
2.6.15 f n , n ( t ) = ( - 1 ) n ( n - 1 ) ! t ( τ - t ) n - 1 f n ( τ ) d τ .
2.6.16 f n , ϕ = ( - 1 ) n 0 f n , n ( t ) ϕ ( n ) ( t ) d t , ϕ 𝒯 ,
2.6.26 lim ε 0 f n , ϕ ε = n ! 0 f n , n ( t ) ( t + z ) n + 1 d t .
f n , j ( t ) being the j th repeated integral of f n ; compare (2.6.15). …
15: 1.4 Calculus of One Variable
Repeated Integrals
16: 36.10 Differential Equations
17: Bibliography H
  • D. R. Hartree (1936) Some properties and applications of the repeated integrals of the error function. Proc. Manchester Lit. Philos. Soc. 80, pp. 85–102.
  • 18: 36.7 Zeros
    Near z = z n , and for small x and y , the modulus | Ψ ( E ) ( x ) | has the symmetry of a lattice with a rhombohedral unit cell that has a mirror plane and an inverse threefold axis whose z and x repeat distances are given by …
    19: 1.6 Vectors and Vector-Valued Functions
    In almost all cases of repeated suffices, we can suppress the summation notation entirely, if it is understood that an implicit sum is to be taken over any repeated suffix. …
    20: Bibliography
  • D. E. Amos (1989) Repeated integrals and derivatives of K Bessel functions. SIAM J. Math. Anal. 20 (1), pp. 169–175.