Relating Pairs of Non-Zero Simple Zeros of Analytic Functions
Relating Pairs of Non-Zero Simple Zeros of Analytic Functions
Edwin G. Schasteen⇤
June 9, 2008
Abstract
We prove a theorem that relates non-zero simple zeros z1 and z2 of two arbitrary analytic functions f and g, respectively.
1
Preliminaries
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 [1].
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 [1].
Z
f (z)dz =
c1
Z
f (z)dz
(1)
c2
Proof of Theorem 1: See pages 129-130 of [1].
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
Edwin G. Schasteen⇤
June 9, 2008
Abstract
We prove a theorem that relates non-zero simple zeros z1 and z2 of two arbitrary analytic functions f and g, respectively.
1
Preliminaries
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 [1].
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 [1].
Z
f (z)dz =
c1
Z
f (z)dz
(1)
c2
Proof of Theorem 1: See pages 129-130 of [1].
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
References: [1] Mathews, John, H. Complex Variables for Mathematics and Engineering Second Edition, Wm. C. Brown Publishers, 1982. [2] Rolf Nevanlinna and V. Paatero, Introduction to Complex Analysis(Reading, Massachusetts: AddisonWesley Publishing Company, 1969), Section 9.7. 20