TEAM PROJECT. Theory on Growth (a) Growth of entire

Chapter 14, Problem 14.4

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TEAM PROJECT. Theory on Growth (a) Growth of entire functions. If fez) is not a constant and is analytic for all (finite) z, and Rand M are any positive real numbers (no matter how large), show that there exist values of z for which Izl > Rand If(z)1 > M. (b) Growth of polynomials. If fez) is a polynomial of degree n > 0 and M is an arbitrary positive real number (no matter how large), show that there exists a positive real number R such that If(z)1 > M for all Izl >R. (c) Exponential function. Show that fez) = eZ has the property characterized in (a) but does not have that characterized in (b). (d) Fundamental theorem of algebra. rf fez) is a po/ynDmial in z, IUlt a constant, then fez) = 0 for at least one value (If z. Prove this, using (a).