For [omega] [member of] [(C - [square root of (-1)]R).sup.r], [[GAMMA].sub.r](s, [omega]) and [S.sub.r](s, [omega]) are meromorphic functions
in s [member of] C.
where j = 1,...,p, l = 1,...,m, and [[phi].sub.i] are real meromorphic functions
. The word 'explicit' means that the variable [y.sub.i](t + [n.sub.i]) does not appear on the right-hand side of the ith equation, i.e.
A meromorphic function
p(z) [??] p(z, [g.sub.2], [g.sub.3]) with double periods 2[v.sub.1], 2[v.sub.2], which satisfies the equation
Suppose that f(z) and g(z) are two nonconstant meromorphic functions
. We define a(z) as a meromorphic function
or a finite complex number.
Let a and b be two distinct finite values and f be a meromorphic function
in the complex plane with finitely many poles.
The restriction of meromorphic function
[[gamma].sub.c] on [C.sub.q] we call Complex Characteristic Function (Complex CF) .
Weierstrass elliptic function p(z) := p(z, [g.sub.2], [g.sub.3]) is a meromorphic function
with double periods 2[[omega].sub.1], 2[[omega].sub.2] and satisfying the equation
Note that [[zeta].sub.E,q](s,x) is a meromorphic function
Let f be a transcendental meromorphic function
in C, all but finitely many of whose zeros are multiple, and let R([not equivalent to] 0) be a rational function.
It is well known that this function extends to a meromorphic function
of [xi] [member of] [V.sup.*.sub.C].
Example in Subsection 3.1 takes the same meromorphic function
as in  to compare the fast IPRM and the G-IPRM proposed in .
For any nonconstant meromorphic function
h(z)wedenoteby S(r, h) any quantity satisfying