Relating Pairs of Non-Zero Simple Zeros of Analytic Functions Edwin G. Schasteen⇤
June 9, 2008
We prove a theorem that relates non-zero simple zeros z1 and z2 of two arbitrary analytic functions f and g, respectively.
Let C denote the set of Complex numbers, and let R denote the set of real numbers. We will be begin by describing some fundamental results from complex analysis that will be used in proving our main lemmas and theorems. For a description of the basics of complex analysis, we refer the reader to the complex analysis text Complex Variables for Mathematics and Engineering Second Edition by John H. Mathews. The following theorems have particular relevance to the theorems we will be proving later in this paper, and will be stated with out proof, but proofs can be found in .
Theorem 1 (Deformation of Contour)(Mathews) If c1 and c2 are simple positively oriented contours with c1 interior to c2 , then for any analytic function f deﬁned in a domain containing both contours, the following equation holds true .
f (z)dz =
Proof of Theorem 1: See pages 129-130 of .
The Deformation Theorem basically tells us that if we have an analytic function f deﬁned on an open region D of the complex plane, then the contour integral of f along a closed contour c about any point z in D is equivalent to the contour integral of f along any other closed contour c0 enclosing that same point z. The Deformation Theorem allows us to shrink a contour about a point z arbitrarily close to that point, and still be guaranteed that the value of the contour integral about that point will be unchanged. This property will be instrumental in the proof of a lemma we will be using in proving our main result that relates all ordered pairs (z0 , z1 ) of non-zero simple zeros, z0 and z1 , of any two arbitrary analytic functions, f and g, each having one of those points as a simple zero. This powerful result is both non-trivial, and counter-intuitive: there is no reason to think right o↵ that all pairs of non-zero simple zeros of analytic functions are related. The result is non-trivial because our result only works for pairs of non-zero simple zeros and does not in general carry over to more than two non-zero simple zeros. All of the statements above will be proven rigorously ⇤ The author wishes to o↵er special thanks to Sean Apple, Dr. Edwin Ford, Ryan Mitchell, and Larry Wiseman for all of their insights and contributions to making this paper possible. Without each one of them, none of what is in this paper, however useful or not, would have been possible.
in this paper. But before this, we wish to describe brieﬂy one case where a more general result does hold; namely, that if the non-zero simple zeros of an analytic function g are closed under multiplication, then the non-zero simple zeros of any other arbitrary analytic function, say h, that is deﬁned on a union of open regions in the complex plane containing all of the non-zero simple zeros of said function g, can be related using a slight modiﬁcation of our main theorem to be proven. All but the last of these statements, too, will be proven rigourously in this paper, as the proof of the last statement is trivial. One particular application of this special case of our main theorem to be proved, is the reduction of the prime factorization problem down to evaluating contour integrals of any number of possible analytic functions over a closed contour. More speciﬁcally, the integral is taken over a closed contour containing information about the prime factors of a product of prime numbers. The product to be factored is contained in the argument of a product of analytic functions, f and g, each of whose only zeros in the complex plane occur at the integers, and the result is a factor of the product of prime numbers. This particular result was the main conclusion obtained via our two year research project...
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