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 books.google.com Let f(u) be an odd elliptic function ; if w is a halfperiod, we must have at the same
time /(«*)=/( w) and f(w)=f( w), since w = w \2w. It is necessary, then,
that f(w) shall be zero or infinite, that is, that w must be a zero or a pole for f(u). 

 books.google.com Let f(u) be an odd elliptic function ; if w is a halfperiod, we must have at the same
time/(w)= /( w) and f(w)=f{w), since w = w + 2 w. It is necessary, then, that
f(w) shall be zero or infinite, that is, that w must be a zero or a pole for f(u). 

 books.google.com 3. Zeros. and. poles. of. meromorphic. functions. In the previous chapters we
have considered the problem of computing all the zeros of an analytic function
that lie in the interior of a Jordan curve. We have seen how the algorithm that we
have ... 

 books.google.com then, for real values of s greater than one, £(s) is equal to Dirichlet's function
However, formula (3) for £(s) "remains ... Thus formula (3) defines a function CW
which is analytic at all points of the complex splane except for a simple pole at s
= 1. 

 books.google.com Here k is system gain, a is ultradamped pole in wplane and q > 0. For simplicity,
take a = 1, k = 1, and frequency response transfer function is H(jω) = 1(jω)q+1.
For small values of frequency H(jω) = 1, and magnitude is thus unity and phase ... 

 books.google.com The corresponding theorem for the poles of an analytic function may be stated as
follows: Theorem VI. The poles of an analytic function are isolated singular points
. If an analytic function f(z) has a pole at any point a, then by Theorem III j, ... 

 books.google.com The determination of the dispersion curves of these modes amounts to finding the
poles of the scattering matrix. The numerical computation of poles of functions of
a complex variable is by no means a simple problem. It cannot be considered ... 

 books.google.com But from formula (3.1) it is clear that the function p(2) is even. Hence, c = 0, i.e., wi
and ^'? are periods of p(2). The function p has double poles at the nodes of the
lattice: and it has no other singular points. Inside the fundamental parallelogram ... 

 books.google.com We recall from Proposition 6.14 that for any polynomial H C C[xi, . ..,xn] the pole (
in t) of the series H(x(t; P(i))) for generic c E Cs is precisely the pole that H\FO has
, viewed as a meromorphic function on the completion of Fc (which is an Abelian
... 

 books.google.com ^2(9 ,s 2;/3.2) 's holomorphic for Re(s) > 0 with exception of s = 1,2,3, where E
%(g', s i; 7s2) achieves a simple pole by the same reason. According to the
above Lemma, the third term \ti\~s+3E%(g',s ;/8,2)C(2s+2)cu+3) 's
holomorphic ... 

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