residue
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11: 25.15 Dirichlet -functions
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►For the principal character , is analytic everywhere except for a simple pole at with residue
, where is Euler’s totient function (§27.2).
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12: 16.11 Asymptotic Expansions
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►It may be observed that represents the sum of the residues of the poles of the integrand in (16.5.1) at , , provided that these poles are all simple, that is, no two of the differ by an integer.
(If this condition is violated, then the definition of has to be modified so that the residues are those associated with the multiple poles.
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13: 22.5 Special Values
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►Table 22.5.1 gives the value of each of the 12 Jacobian elliptic functions, together with its -derivative (or at a pole, the residue), for values of that are integer multiples of , .
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14: 23.2 Definitions and Periodic Properties
15: Bibliography H
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Bernoulli numbers and polynomials via residues.
J. Number Theory 76 (2), pp. 178–193.
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16: 8.2 Definitions and Basic Properties
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►When , is an entire function of , and is meromorphic with simple poles at , , with residue
.
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17: 16.5 Integral Representations and Integrals
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18: 25.2 Definition and Expansions
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►It is a meromorphic function whose only singularity in is a simple pole at , with residue 1.
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19: 25.16 Mathematical Applications
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has a simple pole with residue
() at each odd negative integer , .
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20: Bibliography D
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Irreducibility of certain generalized Bernoulli polynomials belonging to quadratic residue class characters.
J. Number Theory 25 (1), pp. 72–80.
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