About the Project
NIST

residue

AdvancedHelp

(0.000 seconds)

1—10 of 23 matching pages

1: 27.15 Chinese Remainder Theorem
By the Chinese remainder theorem each integer in the data can be uniquely represented by its residues (mod m 1 ), (mod m 2 ), (mod m 3 ), and (mod m 4 ), respectively. Because each residue has no more than five digits, the arithmetic can be performed efficiently on these residues with respect to each of the moduli, yielding answers a 1 ( mod m 1 ) , a 2 ( mod m 2 ) , a 3 ( mod m 3 ) , and a 4 ( mod m 4 ) , where each a j has no more than five digits. …
2: 1.10 Functions of a Complex Variable
The coefficient a - 1 of ( z - z 0 ) - 1 in the Laurent series for f ( z ) is called the residue of f ( z ) at z 0 , and denoted by res z = z 0 [ f ( z ) ] , res z = z 0 [ f ( z ) ] , or (when there is no ambiguity) res [ f ( z ) ] . …
§1.10(iv) Residue Theorem
Suppose that … is analytic in , except for simple poles at z = z n of residue a n . …
3: 2.5 Mellin Transform Methods
2.5.6 I ( x ) = d < z < c res [ x - z f ( 1 - z ) h ( z ) ] + E ( x ) ,
2.5.11 res z = n [ x - z f ( 1 - z ) h ( z ) ] = ( a n ln x + b n ) x - n ,
2.5.35 I j k ( x ) = p j k < z < q j k res [ - x - z G j k ( z ) ] + E j k ( x ) ,
2.5.46 res z = k [ - ζ z - 1 Γ ( 1 - z ) π csc ( π z ) ] = ( - ln ζ + ψ ( k ) ) ζ k - 1 ( k - 1 ) ! ,
2.5.47 res z = 1 [ - ζ z - 1 Γ ( 1 - z ) h 2 ( z ) ] = ( - ln ζ - γ ) - h 1 ( 1 ) ,
4: 5.2 Definitions
It is a meromorphic function with no zeros, and with simple poles of residue ( - 1 ) n / n ! at z = - n . … ψ ( z ) is meromorphic with simple poles of residue - 1 at z = - n . …
5: 5.19 Mathematical Applications
By translating the contour parallel to itself and summing the residues of the integrand, asymptotic expansions of f ( z ) for large | z | , or small | z | , can be obtained complete with an integral representation of the error term. …
6: 27.2 Functions
Such a set is a reduced residue system modulo n . …
7: Mathematical Introduction
( a , b ] or [ a , b ) half-closed intervals.
res residue.
8: Bibliography V
  • J. F. Van Diejen and V. P. Spiridonov (2001) Modular hypergeometric residue sums of elliptic Selberg integrals. Lett. Math. Phys. 58 (3), pp. 223–238.
  • 9: 2.3 Integrals of a Real Variable
    2.3.18 b s = 1 μ res t = a [ q ( t ) ( p ( t ) - p ( a ) ) ( λ + s ) / μ ] , s = 0 , 1 , 2 , .
    10: 25.15 Dirichlet L -functions
    For the principal character χ 1 ( mod k ) , L ( s , χ 1 ) is analytic everywhere except for a simple pole at s = 1 with residue ϕ ( k ) / k , where ϕ ( k ) is Euler’s totient function (§27.2). …