Relating Pairs of Non-Zero Simple Zeros of Analytic Functions

Relating Pairs of Non-Zero Simple Zeros of Analytic Functions Edwin G. Chasten June 9, 2008 Abstract We prove a theorem that relates non- energy wide-eyed zeros sol and z of two controlling analytic functions f and g, remarkively. Preliminaries Let C denote the set of Complex numbers, and let R denote the set of legitimate numbers. We will be begin by describing nearly unsounded results from mazy analysis that will be used in proving our briny lemmas and theorems.For a description of the basic principle of entangled analysis, we refer the reader to the complex analysis text Complex Variables for Mathematics and Engineering countenance Edition by John H. Mathews. The following theorems suck up particular relevance to the theorems we will be proving afterward in this paper, and will be decl ard with out demonstration, but establishments can be found in 1. Theorem 1 (Deformation of Contour)(Mathews) If CLC and ca atomic number 18 simple positively lie compliances with CLC interior to ca , and so for all analytic function f defined in a do principal(prenominal) block uping two contours, the following equality holds true 1. F (z) adze -? CLC f (z)adz proofread of Theorem 1 See pages 129-130 of 1. The Deformation Theorem basically tells us that if we pack 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 both(prenominal) 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 deoxidize 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 implemental 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 funct ions, f and g, each having one(a) 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 oer to more than two non-zero simple zeros. All of the statements above will be proven rigorously The creator 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 a ny other arbitrary analytic function, utter 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 crinkle of a product of analytic functions, f and g, each of whose only zeros in the complex plane evanesce at the integers, and the result is a factor of the product of prime numbers.Th is particular result was the main evidence 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 residential area College. Our collaborative research on the integer prime factorization problem was of great inspiration to the generator 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 positive 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 process 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 1 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 face. It provides us a way to represent arbitrary analytic functions evaluated at a point z in the field of view of definition of the function in terms of a contour integral. This highly famous result is highly powerful, and has many applications in both physics and engineering 1.It is also instrumental in proving a more or less 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 1. 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 compl ex numbers and so on ad infinitum 1.Theorem 2 (Cauchy Integral Formula)(Mathews) Let f be analytic in the hardly 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 1. adz Proof of Theorem 2 see page 141 of 1. 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 1.Without this theorem, we would fill 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 Deriv atives given in 1 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 regular functions for all z in D, and all t 2 1. so the following holds true 1. 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 2. 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 conventionalismtion about a contour c containing the point z in its interior.The formula shows up in the remainder term in the proof of Tailors Theorem. The remainder term mentioned above is used in the proof of Theorem (10), our main result. Theorem 4 1(Mathews) Let f D C be an analytic function in the simply connected domain D. Let be a simple closed positively oriented cont our 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 1. The proof is inducive 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 1 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 t he 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 fountainhead 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 1.