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 CLC and ca are simple positively oriented contours with CLC interior to ca , then for any analytic function f defined in a domain containing both contours, the following equation holds true . F (z)adz -? CLC f (z)adz 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 off long a closed contour c about any point z in D is equivalent to the contour integral of f along any other closed contour co 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 (zoo , sol ) of non-zero simple zeros, zoo and sol , 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 owe 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 proper special thanks to Sean Apple, DRP. Edwin Ford, Ryan Mitchell, and Larry Washman 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 rigorously in this paper, as the proof of he 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 consisting of the following researchers: Sean Apple, DRP. Edwin Ford, Ryan Mitchell, and Larry Washman, math instructor at Pierce Community College. Our collaborative research on the integer prime factorization problem was of great inspiration to the author in the formation of the generalization that is the main theorem of this paper.
This main theorem, itself, is a generalization of some machinery we had together developed to reduce the prime factorization problem to evaluating contour integrals of the product f two specially chosen functions in the complex plane during the two year research project. The author wishes to thank Sean Apple, DRP. Edwin Ford, Ryan Mitchell, and Larry Washman, for their inspiration and help in making this generalization possible, for without them, none of this, however useful or not, would have been discovered at this time.
For the following discussion, see page 113 of  for a formal definition of a contour. Now we shall discuss some more theorems that will be instrumental in proving our main results. The following theorem is called Cauchy Integral Formula. It provides us a way to represent arbitrary analytic functions evaluated at a point z in the domain of definition of the function in terms of a contour integral. This highly famous result is extremely powerful, and has many applications in both physics and engineering .
It is also instrumental in proving a most counter-intuitive result: that if a function f is determinable on an open subset of the complex plane (I. E. If f is analytic on an open subset of the complex plane), then f has derivatives of all orders on that set . In other words, if a function f has a first derivative on an open subset f complex numbers, then it has a second derivative defined on the same open subset of complex numbers, and it has a third derivative defined on the same open subset of complex numbers and so on ad infinitum .
Theorem 2 (Cauchy Integral Formula)(Mathews) Let f be analytic in the simply connected domain D, and let c be a simple closed positively oriented contour that lies in D. If zoo is a point that lies interior to c, then the following holds true . adz Proof of Theorem 2: see page 141 of . The following theorem is called Leibniz Rule and along with Cauchy Integral Formula is instrumental in proving what is known as Cauchy Integral Formula for Derivatives, which has as a corollary, that functions that are analytic on a simply connected domain D, have derivatives of all orders on that same set .
Without this theorem, we would need much stronger assumptions in the premise of our theorem relating pairs of non-zero simple zeros of analytic functions. Although we shall not use Leibniz rule directly in any of our proofs, Leibniz rule together with Cauchy Integral Formula form the back-bone of the machinery in the proof of Cauchy Integral Formula for Derivatives given in  on page 144, which we shall only outline. 2 Theorem 3 (Leibniz Rule)(Mathews) Let D be a simply connected domain, and let I : a t 0 b be an interval of real numbers.
Let f (z, t) and its partial derivative fez (z, t) with respect to z be continuous functions for all z in D, and all t 2 1. Then the following holds true . B f (z, t)dot fez (z, t)dot is analytic for z 2 D, and Proof of Theorem 3: The proof is given in . The following Theorem is called Cauchy Integral Formula for derivatives and allows one to express the derivative of a function f at a point z in the domain off by a onto integral formula about a contour c containing the point z in its interior.
The formula shows up in the remainder term in the proof of Tailor's Theorem. The remainder term mentioned above is used in the proof of Theorem (10), our main result. Theorem 4 (Mathews) Let f : D ! C be an analytic function in the simply connected domain D. Let be a simple closed positively oriented contour that is contained in D. If z is a point interior to c, then n! Ads z)n+l Proof of Theorem 4: We give here a sketch of the proof appearing in . The proof is inductive and starts with the parameterization C : s = s(t) ND Ads = s (t)dot for a 0 t 0 b.
Then Cauchy Integral formula is used to rewrite f in the form O f (s(t))so (t) dot s(t) z The proof then notes that the integrands in (B) are functions of z and t and the f and the partial derivative off with respect to z, fez , are derived and then Leibniz rule is applied to establish the base case for n = 1. Then induction is applied to prove the general formula. The main point of this is Corollary (5. 1) in  on page 144, which states that if a function f is analytic in a domain D, then the function has derivatives 3 of all orders in D, and these derivatives are analytic in D.
Without this corollary, we could not relate the non-zero simple zeros of analytic functions as stated in Theorem (10); instead, the best we could do is to relate the non-zero simple zeros of functions whose second derivative exists on the intersection of the domains of the functions that contain the pair of non-zero simple zeros of the pair of given functions. But with Corollary (5. 1), we need only assume analyticity of the functions in question at the non-zero simple zeros, which significantly strengthens the results of our paper.
Below we will give the definition of what is known in complex and real analysis as a ere of an analytic function f of a given order k, where k is a non-negative integer. What the order of a zero z tells us is how many of the derivatives of the function f are zero at z in addition to f itself. What is known is that if two functions, f and g, have a zero of order k and m, respectively, at some point zoo in the complex numbers, then the product of the two function f and g, denoted f g, will have a zero of order k + m at the point zoo .