ISSN: 2250-3005 - ijcer

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International Journal Of Computational Engineering Research (ijceronline.com) Vol. 2 Issue. 8Thus,j1z1j1jcQ( z 1) c0,But this was supposed to be our minimum. Thus , a contradiction. Hence proved06. Proof of the Fundemental Theorem via Radius of convergenceWe now prove the Fundamental theorem of Algebra: As always, p(z) is a non constant polynomial. Consider1f ( z) b0 b1z ...p(z)andnp( z) anz ... a , a0Lemma. c, r Csuch that00kbk cr for infinitely many k.Now, 1= p(z)f(z). Thus, a 0 b 0 = 1. This is our basic step .Assume we have some coefficient such that │b k │ > cr k . We claimwe can always find another .Suppose there are no more .Then the coefficient of z k+n in p(z)f(z) isa0bkn a1bkn1... anbkThus, as we havea0jbj cr in this range ,we have the coefficient satisfiesn n10r a1r ...an1ranfrmin 1,ana0 ... an1This will give thatbkab0 k n...n1k 1a anbbka0 b ... knn1k1anabcrkfor sufficiently s mall.Let z = 1 , Thenrbk kbkz krThis is true for infinitely many k, hence theand its power series converges everywhere.cpower series diverges, contradicting the assumption that function is analytic7. Proof Of The Fundamental Theorem Via Picard’s TheoremStatement: If there are two distinct points that are not in the image of an entire function p(z) (ie. z1 z such2that for allz C, p(z) z orz ),then p(z) is constant.We now prove the Fundamental Theorem of Algebra;12||Issn 2250-3005(online)|| ||December|| 2012 Page 307
International Journal Of Computational Engineering Research (ijceronline.com) Vol. 2 Issue. 8Let p(z) be a non-constant polynomial, and assume p(z) is never 0.Claim:- If p(z) is as above , p(z) does not take on one of the variablek1 fork NProof: Assume not. Thus, z kCsuch that 1p( z k) . If we take a circle D centered at the origin withksufficiently large radius, then │p(z) │ > 1 for all z outside D. Thus each z 1 D , we have a convergent subsequence. Thuswe have'n z .butzip ( z' ) lim p(z ) 0 .niniThus there must be some k such that 1p( z) . Since there are two distinct values not in range of p, by picard‟skTheorem it is now constant. This contradicts our assumption that p(z) is non-constant. Therefore, p(z 0 ) = 0 for some z 0 .Remark:- One can use a finite or countable versions of picards. Rather than missing just two points, we can modify the aboveto work if Picard instead stated that if we miss finitely many (or even countably) points, we are constant. Just look at the1method above, givesk . We can then find another larger one, say .And so on. We can even get uncountably manyk21such points by looking at numbers such as2(using now the transcendence of C is 1).18. Proof Of The Fundamental Theorem Via Cauchy’s Integral TheoremStatement:- Let f(z) be analytic inside and on the boundary of some region C. Thenf ( z) 0cWe now prove the Fundamental Theorem of Algebra.Proof: Without loss of generality let p(z) be a non- constant polynomial and assume p(z) = 0. Forp(z)( .Therefore, p(z) doesn‟t change signs for2d 0p(2cos)0 R (Otherwise, consider p z)pzz R.assumez R , or by the Intermediate Value Theorem it would have a zero.This follows from our assumption that p(z) is of constant signs for real argument, bounded above from 0. This integral equalsthe contour integral1iz 1If z ≠ 0, Q(z) ≠ 0dzzp(z1i1 z )z 1If z =0, then Q(z) ≠ 0 SinceP(z + z -1 ) = a n (z + z -1 )+ . . .z n p(z + z -1 ) = z n (…a n z -n ) + . . .= a n + z( . . . )Thus, Q(z) = a n , which is non-zero . Hence, Q(z) ≠ 0 ,ConsequentlyFormula1i1z nQ(z)z 1is analytic. By the Cauchy Integraln1z 0. Thus, a contradiction!Q(z)n1zQ(z)||Issn 2250-3005(online)|| ||December|| 2012 Page 308
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