On the fundamental theorem of algebra

Transcription

1 Atlanta University Center W. Woodruff Library, Atlanta University Center ETD Collection for AUC Robert W. Woodruff Library On the fundamental theorem of algebra Riichie Dean Williams Atlanta University Follow this and additional works at: Part of the Physical Sciences and Mathematics Commons Recommended Citation Williams, Riichie Dean, "On the fundamental theorem of algebra" (1960). ETD Collection for AUC Robert W. Woodruff Library. Paper 799. This Thesis is brought to you for free and open access by W. Woodruff Library, Atlanta University Center. It has been accepted for inclusion in ETD Collection for AUC Robert W. Woodruff Library by an authorized administrator of W. Woodruff Library, Atlanta University Center. For more information, please contact

2 ON THE PUNDI~NENTAL TH~R~ OF ALGEBRA A THESIS SUBMITTED TO THE FACULTY OF ATLANTA UNIVERSITY Th PARTIAL FULFILU4~T OF THE REQUIREtVIENTS FOR THE DEGREE OF MASTER OF SCIENCE BY RICHIE DEAN WILIJAMS DEPARTMENT OF MATHEMAUCS ATLANTA, GEORGIA AUGUST 1960

3 To my mother, Mrs. Alice B. Williams 11

4 2.11 ACKNOWLEDGMENT I wish to express my thanks to the National Science Foundation for providing the opportunity for me to advance my education; thus, leading to the completion of this thesis, My most humble gratitude is extended to Dr. Lonnie Cross for his rigorous and critical suggestions, and special thanks to Dr. S. C. Saxena for his very helpful and timely suggestions. R. D. W.

5 iv TABLE OF CONTENTS Chapter I, INTRODUCTION ~.. :~ II. MATH~iATIGAI3 TOOLS ~ Page 2 1. Equations TheSiinpleClosedCurve..., , Cauchyts IntegralTheorem...,..., )i. ~n Ebctension of Cauchy s Integral Theorem The Residue Theorern The Zeros ofafunction..., Rouche s Theorem III. PROOF OF THE FUNDP~ENTAL THEOR~ BIBLIOGRAPHY

6 CHAPTER I INTRODUCTION This is a study on the fundamental theorem of algebra which belongs to the field of analysis. Hence, I shall proceed to give a proof of the fundamental theorem of algebra by means of analysis. Some of the powerful mathematical tools of anaj.ysis are applied. These tools include Cauchy s integral, theorem, the residue theorem, and the zeros of an analytic function inside the unit circle and Rouche s theorem. AU material not pertinent to this discussion has been deleted in order to make for s:implicity in presentation and understanding. As a result, this proof should provide an interesting study on the fundamental theorem of algebra. Several proofs of this theorem have been given. However, the first complete proof that solutions exist was given by Gauss in his doctoral thesis in

7 CHAPTER II M~LTH~ATICAL TOOLS 2-1. Equations The fundamental theorem of algebra toils us of the existence of solutions for all algebraic equations with complex coefficients. That is to say, the polynomial equation, az~+az~1+a2z~2+, where n is positive, a0 is not zero and a0, a.~, a2,..., a~ are complex numbers, has at least one solution in the field of complex numbers. Since the solution of these equations belong to the field of complex numbers, let us consider the geometric structure of complex numbers. Let z = x + i y. Then r = j z j = 1/ x2 + y2 is called the modulus (or absolute value) of z. Consider Figure 1. Set e tan~ ~. y*~y Then x = r cos 9 and y = r sin 8. 8 is called the Figure 1 argument of z. Now observe z = x + i y y(cos e + i sin e). This form is the polar form of z in which ~ The Simple Closed Curve Let x = x (~) and y = y (t) be continuous on These functions may be thought of as the parametric ~oc) Figure 2 representation of the curve C (Fig. 2)j ~:i~iing~: ~póints FP(x ( V, y (~)a~d Q Cx (,e), y () ). 2

8 3 Set z (t) = x (t) + iy Ct). Then our representation is simpler. Thus z = z (t), on, represents the curve C, If z (ot) = z (,8), then our curve is said to be a closed curve. If z (t,~) ~ z Ct2) for all t1 ~ t2 in ~ but z (s.) z (,.d), then our curve is called a simple closed curve. Sucti a curve is called a Jordan curve. Any simply closed curve (or a Jordan curve) divides the plane into exactly two regions. This theorem is called Jordan s curve theorem.1 A Jordan curve has both an inside or bounded region (cafled the interior region of the curve), and an outside or unbounded region (called the exterior region of the curve). For example, the circle ~ z I = i... divides the plane into two separate regions: j z I ~. t~which represents the interior and z I ~ which represents the exterior Cauchy s Integral Theorem Let f be regular in a simply connected region G.2 Let V be any closed path (not necessarily simple) lying in G. Then (z) d~z 0 a-according to Konrad Knopp, Theory of ~rnctions, Part I, page 15, the proof of Jordan s curve theorem can be found in G. N. Watson, Complex Integration and Cauchy s Theorem, Cambridge Tract No. 15, l9llt., Chapter I. region is said to be simply connected if every simple closed path lying entirely within the region contains only points o± the region itself,

9 /it J _ 14fl, n = 1, 2,... Proof: Part I, 14 Let ~ be a triangle~contajnedjn G. Divide T into 14 congruent parts T ~, T~, T~, T~, by passing lines I, to the sides of T. We want each triangle traversed in the positive mathematical sense. In Figure 3 note: S~.f~ ~ + 5- T~ + 5T ~ 1 T~ canceil out each other where the effect of inte grating back and forth on the subsidiax~r paths is nullified. There exists at least one of these triangles ~ call it such that (z) dz/ * (z) dz/ i. e. /f f (z) d ~ ~ d We repeat the process for T1. Thus we get some T2 such that )ST1f (z)dz)~ 14 /ST~f (z)dz/ and hence /5T f (z) d ~ 142 /5 f (z) d and the Area T = ~ the Area of T1 = ~ Area of T. By induction we get a sequence.{ T} of triangles such that~ ~T 1 T0 = T. The area of T~ the area of T~1 and j -c j s ~ /~ / ~. ~ 4~ I~ I ~... or in one state ent

10 Hence the nested sequenôe defines a unique complex number S such that z0e T~ for au n. Thus ~ G since T~C. G for each n. Since f is regllj.ar in G, for any ~ 7i~ ~ (() such that I f(z).~f(z0) f(z0) ~ z z0 whenever / z Set z) = f (z) f (z0) f (z0). z zo Then f (z) = f (zn) + f (z0) (z z0) +~(z, z) (z z0) There exists an index no such that T~+i ~- /V(z0, S) L z / z_z0i43 for ail n ~ no. Now consider ff(z)dz= f(z0)dz+ zf1(z)dz _90f(z0)dz+f ~L(zo,z) (z z0)dz=o O+O ~L z) (z - z0) d z. /I f(z)dz/~l_.~.p. Pfl where P = perimeter of T and / z z J is at most ~ P (since z and n n 0 are points of one and the same triangle). Recall P1 = 4 periineter~ of T. Thus P~ ~ P~_~...

11 Hence 6 = -~ / JTJ_~/JTfdz/~ so that / ff(z)dz/~e But ( is arbitrary. Hence f (z) d z = 0 Part II. An arbitrary polygon Q. E. B. Using the fact that every polygon can be triagulated, we get f (z) d= (f~ ft2 + + JT)~ (z) dz = = 0, where T1, T2,..., T~ are the triangles obtained by triangulating ~. Part III. An Arbitrary Closed Path. We use the fact that any closed path can be approximated as close 1 A Proof to Part II is given by Konrad Knopp, ~eory of Thinctions, Part I, l9l~5, pp

12 as we please by closed polygons L1.. Q. E. D. An Eb~tension of Cauchyts Integral Theorem2 ~ be any simple closed path in 0 and let ~, f~,... be similarly oriented simple closed paths contained in G and each of which is exterior to the other and, all of which lie in 0. Then f (z) d z + +f Jf (z) d z Residue Theorem Let f be regular and single-valued in a region 0, except for a finite number of poles also let C be a simple closed positive oriented path contained in G and not passing through any pole of f (z). Then 1 ( f (z) d z = sum of the residues of f (z) at its poles 2lri ) C inside C. Proof: Let z1, z2,.., Z~ be the finite number of poles of f (z) inside C. Let C~ be circles about as centers such that C.~ ( \C~ = 0 if i ~ (i, ~ = 1, 2, 3,.., k). By an extension of Cauchy s integral theorem~ proof to Part III is given by Konrad Knopp, Theory of Part I, 191i.5, pp. S2 5l~. unctions, suggested method of proof is given in Konrad Knopp, TheOi~T of ~mctiq~, Part I, pp. 5) See Konrad Knopp, Theory o Funq~ions, Part I, pp. LJ.7 5)4.

13 we have 8 1 1f(z)dz= 1 LIf(z)dz+ (f (z)dz+, 2lri 27Ti Jc2. J I (z) d z] Q.E.D The Zeros of a Function A value of z for which I (z) = 0 is called a zero of I (z). Theorem: Let I be meromorphic in a bounded region G. Let I be regular on the boundary of C of G and I (z) ~ o on C, Then ~ N P= 1 ( f1(z)dz,whoren=thenumber 2,ri of zeros of f inside C4 P = the number of poles of I inside C. Proof: Let a1, a2,,,, z~ be the zeros of I (a) inside C and let c~ 1 k be their respective multiplicities, Let ~ f2.., in be the poles of f (a) inside C and let,~ /~2 be their respective orders. Then observe that I (a) (a P. (a), ~mere P (a) ~o for a = a, j 1, 2, and I (z) 1 Q1 (a), where Q1 (a) is regular at a., i = 1, (a~ - 2,,,, in.

14 Moreover 9 f ~ (a) =~ + P (a), j = 1, 2,..., f (~T a a. P. (a) J and ~ (a) + Q~ (a), i = 1, 2,.. Q(~ Thus the function f~ (a) has inside C, simple poles at a1, a2,., f (a) Zk~ ~ l~ ~2 * ~ Hence by the residue theorera k m 1 2 ~ ic. f (~) j1 J i1 ~- since N =~ and P =~,8.. Suppose f is regular throughout G. Q. E. D. Then ~becoraes N=l f (z)dz. 277 i f(z) Set d log f (a) I / (a) d a. f (~)_ Then N 1 ( d log f (a) 1 5 log f (a), 2~ri 2~~i 0: where log f (a) reads variation of log I (a) around the closed contour C.

15 But we know that: 10 log f (z) = log f (z) I and i arg f (z) Hence N 1 fl~ log Jf (z)i +~iargf (z)j 2lri i.~c 1_ ~ argf (z), - ~7t C since ~ log g4 f (z) I 0. Therefore the number of zeros of f (z) interior to C, counted ~dth their imiltiplicities is given by N= 1 ~ argf (z). 27r C (This is lmown ao tnc principle of the ar-gumeiit.) 2 7. Rouche s Theorem Let G be a sbraply connected region bounded by a simple closed curve C. Let f and g be regular in G LI C, f (z) ~ o on ~, and f (z) j > jg (z) ~ on C. Then f (z) and f (z) + g (z) have the same number of zeros inside C, We observe that neither f (z) nor f (z) + g (z) has a zero on C, and so, if is the number of zeros of f (z) and Nf + g is the number of zeros of (z) + g (z) 2WNf~cJargf(z) 2~TNf 4~arg(f(z)+g(z)) Wewanttoshow2~rN f+g f

16 U But observe 2?TNf + g arg (f (z) + g (z) ~ arg (z) (1 + g (z)] = ~ [are i~ (z) + arg (1 + g (z) C f(z) J ~ argf (z)+~k~arg (1+g_(z)) C f(z) 2 ~?Nf + U arg (1 + g (z) ) C f(z) We must now show 4 arg (1 + g (z) ) = 0 f(zj Since g (z) ~,/ on C, we have that 1 + g (z) lies inside the f(zj f(z) circle / 1 w / 4 1 and hence as we go around the curve C the path traced by w 1 + ~(z) cannot encircle the origin in the w plane. Hence fczy 4 arg(l+g(z))=o,,. N~ Nf. C Q. E. D.

17 CHAPTER UI PROOF OF THE FUNDM~ENTAL THEOR~1 equation, The fundamental theorem of algebra states that every algebraic a0 ~ + ~n~l + a2 ~n2 +,, + = 0 where n is positive, a0 is not zero, and a0, a)~ a2,.., a~ are complex numbers, always has at least one complex root, Let f (z) = a z1 (a ~ o) and g (z) a1 ~ +,.. + Let C be the circle z ~ ~Y) Y ~ 0. Choose Y so that f (z) ~ ~ g (z) I on C. Let z = r ewe, ~there eig cos 9 i sin 9. f (z) = a0 rt~~ g (z) = a1 ~ ~- e~ (n~) ~ + a~ rn.~2 0j(~2) e +,~e + a~. 12

18 Now observe that / a0 r1~ e~~i /a, r ~- e~ (n i) ~ + + ~ I 13 a0 / r~ a1 1 + /a2j 1 + ja~j ~ j~j 2 Since the expression in [ ] is a poj~momial in 1 and since ovor~r po]~jnomiaj. is oontinuouc, there exists r0 ouch that the expression in ( ]~ for all r > r0. Hence / f (z) / /g (z) / ~ ~ / a0 / r1~ for all r > r0. Thus taking any one of these r we have /f (z)i ~./g (z)~ on C:/zj Now it is oloar that the hypothesis of Rouche s theorem are satisfied, Hence the number of zeros of f (z) and f (z) g (z) interior to C is the same. But it is clear that the number of zeros of f (z) inside C is n, Since f (z) + g (z) ~ a~ ~ + a1 z~~1 +,, we have that a0 zr~ + a1 z~ ~ +,, a~ 0 has co~1ex roots, Q. E. B.

19 BThLIOGRAPHY Birkhoff, Garrett and MacLane, Saunders, A Survey of Modern Algebra, New York, The Macmillan Company, Churchill, Ruel V., Introduction to Complex Variables and Applications, New York, NcGraw-.Hiil Book Company, Incorporated, 191i8. Courant, Richard and Robbins, Herbert, Nhat Is Mathematics, 8th reprint, New York, Oxford University Press, Dickson, Leonard Eugene, First Course in the Theory of Equations, New York, John Wiley and Sons, Incorporated, Hardy, G. H., A Course of Pure Mathematics, 10th edition (reprint), Cambridge, University Press, Knopp, Konrad, Theory of Functions, Part I, New York, Dover Publications, Knopp, Konrad, Theory of Functions, Part II, New York, Dover Publications, Marden, Morris, The Geometry of the Zeros of a Polynomial in a Complex Variable, New York, knerican Mathematical Society, Meserv~, Bruce E., Fundamental Concepts of Geometry, Reading, Massachusetts, Addison-Wesley Publishing Company-, Phillips, E. G., Functions of a Complex Variable with Applications, 8th edition (reprinted), New York: Inter science Publishers, Incorporated, Uspensky, J. V., Theory of Equations, New York, McGraw-Hill Book Compar~y, Incorporated, ~I;4