ISSN: 2250-3005 - ijcer

International Journal Of Computational Engineering Research (ijceronline.com) Vol. 2 Issue. 8Using in (I) we getN2N112i1c'dzN1 21' N1 2 iSince we have seen that 1c1 dz (II) and so binomial expansion of 1thus obtained is uniformly convergent and hence term by term integration is permissible, Hence' 1'2 31 dz 1...dzcc1 is possible and binomial expansion' 1' '' 21 dz dz dz dz+ …cc cc (III)The function f and g both are analytic within and on C and f(z) ≠ 0 for any point on C . Hence isanalytic and non–zero for any point on C. Therefore and it‟s all derivatives are analyticgf By caucly‟s integral theorem, eachc' ' 1 dz 0In this event (2) takes the formintegral on R.H.S of (3) vanishes consequently.N 2 - N 1 =0 or N 1 - N 23. Proof of Fundamental Theorem of AlgebraConsider the polynomiala 0 + a 1 z + a 2 z 2 + . . . + a n z nsuch that a n ≠ 0Take f(z) = a n z ng(z) = a 0 + a 1 z + . . . + a n-1 z n-1Let C be a circle │ z│= r where r > 1.Then│g(z) │ │ a 0 │ + │ a 1 │r + │ a 2 │r 2 +. . . +│a n-1 │r n-1│g(z) │ │ a 0 │ r n-1 +│ a 1 │ r n-1 +│ a 2 │ r n-1 ….+ │ a n-1 │r n-1│g(z) │ [│a 0 │+ │a 1 │ +│a 2 │ + . . . +│a n-1 │]r n-1But│f(z) │= │a n z n │= │a n │r n g(z)a0 a1 a2 ... an1g z)f ( z)f ( z)aa0 a1 a2 ... a r( n1nnrnarn1||Issn 2250-3005(online)|| ||December|| 2012 Page 303