Meromorphic functions

For the sake of simplicity, we will consider an analytic function / = f(z) of a single complex variable z. It is well known that except for the integer functions and the meromorphic functions defined in the entire complex plane, such an analytic function usually has a natural domain restricted by a boundary of singularities, over which it cannot be analytically continued. This means that even in the case of complex scaling v = v(rj)—for certain values of 11—the transformation may take the variable rjz outside the domain of analyticity, and the operation is then meaningless. As an example, we may consider the analytic functions / = f(z) which are defined only within the unit circle z < 1, with this circle as natural boundary, and it is then evident that the complex scaling is meaningful only for 7 < 1. 

Many of the results on unipotent and solvable groups were first introduced not for structural studies but for use in differential algebra. We can at least sketch one of the main applications. For simplicity we consider only fields F of meromorphic functions on regions in C. We call F a differential field if it is mapped into itself by differentiation. An extension L of such an F is a Picard- Vessiot extension if it is the smallest differential field which contains F together with n independent solutions yt of a given linear differential equation... 

By definition, we represent this analytic continuation by the same symbol / (a). It will be shown that /(a) is a meromorphic function which has poles for integer values of a(a = 1,2,, . . ) and that, forn+l[Pg.859]

Nevertheless, it must be noted that in this way, we do not obtain a completely renormalized theory for all values of the space dimension d. In the regularized theory, the contributions of the diagrams remain meromorphic functions of d. Moreover, poles appear for physical values of d, in particular for d = 3. 

Instead of constructing L-periodic meromorphic functions / on O, one seeks... 

Now if g > 2, most complex tori C9 /L have no non-constant meromorphic functions on them at all, and are not algebraic varieties, and do not carry any but trivial ea s27. In the case of a curve C, however, special things happen let s look for bilinear forms as candidates for B. We saw above that on R C) one has a positive definite Hermitian form ... 

Theorem. The existence of a positive definite Hermitian H on C9 and an integral skew-symmetric E on L satisfying E = Im H is necessary and sufficient for a complex torus C9 /L to carry g algebraically independent meromorphic functions and if it has such functions, it admits an embedding into Pn, some n, hence is a projective variety29. 

The following theorem from the theory of meromorphic functions applies ... 

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