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1MTH302 Chapter 2Modulus of a Differentiable Functionand the Argument PrincipleEntire functions and polynomials (Ref: Section 1.3, Unit 2)Theorem (Ref: Th 1.4, Unit 2)Let f be an entire function and suppose there exists k > 0, R > 0 and positive integer n suchthat | f (z) | ? k |z|n whenever |z| > R. Then f is a polynomial of degree ? n.Proof Let 20 1 20( ) , j j j jjf z a a z a z a z a z z C? ?? ? ? ? ? ? ? ? ?be the Taylor series of f at 0.Let m be an integer > n. Let M(r) = sup {| f (z) | : |z| = r}.Then Cauchy integral formula implies1 1(0, ) (0, )111 ( ) 1 | ( ) || | | |2 2 | |1 ( )[2 ]2 1[2 ]20 for asj j jC r C rj njj nf z f za dz dzi z zM rrrkrrrkj n rr? ?????? ????? ?? ?? ? ? ? ?? ?Hence 0aj ? for all j > n.2Using the same approach, we also have the following theorems and corollaries. Detailedproof are available in the reference for each result.Theorem (Ref: Th 1.5, Unit 2)Let p(z) = a0 + a1z + . . . + anzn where an ? 0. Then p(z)/(anzn) ? 1 as z ? ?.Corollary (Ref: Cor 1.6, Unit 2)Let p(z) = a0 + a1z + . . . + anzn where an ? 0. If n ? 1, then 1/p(z) ? 0 as z ? ?.Theorem (Ref: Th 1.7, Unit 2)Every non-constant complex polynomial has zeros.ProofIf p(z) is a polynomial without zeros, then 1/ p(z) is an entire function.Clearly11 ( )0( )p zas zzp z z? ? ? ?Now 1/ p(z) is an entire function implies that it has a Taylor’s series expansion about 0 givenby01,( )jjja z z Cp z? ?? ? ?where11 ( ) ( ) 1| | [2 ] , ( ) sup{| | | } (*)j 2 ( ) j jM r M ra r M r z rr r p z?? ?? ? ? ?Since 1 ( ) 0 as 0 as( )M rz rzp z r? ? ? ? ? ? ?(*) implies that aj ? 0for all j =1, 2, 3, …Thus p(z) is a constant if it does not have any zero.3Using mathematical induction, we have the following corollary which is commonly called thefundamental theorem of algebra.Corollary (Ref: Cor 1.8 (first one), Unit 2)If p is a complex polynomial of degree n ? 1, then there exists unique c, a , a , . . . , a 1 2 nsuch that p(z) = c(z – a1)(z – a2) . . . (z – an).Maximum modulus principleTheorem (Maximum modulus principle)Let f be analytic in a bounded, open and co

nnected set D (= bounded domain) in C. Supposef is also continuous on the boundary of D. Thensup{| f (z) | : z in D and its boundary}is attained at a boundary point of D; that is, the maximum value of | f (z) | occurs on theboundary of D.CorollaryLet f be analytic in a bounded, open and connected set D (= bounded domain) in C. Supposef is also continuous on the boundary of D and f is non-zero in D. Theninf{| f (z) | : z in D} is attained at a boundary point of D; that is, the minimum value of | f (z) |occurs on the boundary of D.Corollary (Ref: Cor 2.4, Unit 2)If f is analytic in a bounded domain D (= bounded, open and connected D) and | f (z) | attainsits maximum value at a point z0 in D, then f is constant in D.CorollaryLet f be analytic within a simple closed contour ? and continuous on ?. Thensup{| f (z) | : z is within or on ?} is attained at a point of ?.4Theorem (Ref: Th 1.1, Unit 2)Let G be a connected open subset of C. Let f be analytic in G. Then given any point a in Gand any complex constant ?, there exists a positive real number r (depending on a) such thatf z z a r ( ) for all 0 | | ? ? ? ? ? .Theorem (Liouville’s theorem)If f is entire and bounded in the complex plane, then f (z) is constant throughout the plane.Meromorphic functionsDefinitionA function f is said to be meromorphic in a domain D if at every point of D, f is eitheranalytic or has a pole (or both).The argument principleLet f be analytic and non-zero at each point of a counter-wise simple closed curve ? andmeromorphic inside ?. Then01 ( )( ) ( )2 ( ) pf zdz N f N f?i f z ??? ? ? ,whereN f 0( ) is the number of zeros (counting multiplicities) of f inside ? and N f p( ) isthe number of poles of f inside ?CorollaryLet f be analytic inside a counter-wise simple closed curve ? and non-zero at each point onthe curve ?. Then01 ( )( )2 ( )f zdz N f?i f z ??? ? ,whereN f 0( ) is the number of zeros (counting multiplicities) of f inside ?.5Theorem (Rouche’s theorem) (Ref: Th 1.3, Unit 3)Let f and h be both analytic inside and on a simple closed curve ? such that | h(z) |