7. Let f be an entire function. Suppose there exists € >0 such that f(2) >...
Suppose fis an entire function such that there is a number M such that Re(f(z)2 - Imf(z))2 s M for all z. Prove that f must bie Hint: Compare to exercises #8: if f and g are both entire, then so are f-g and gof. Find an appropriate g so that you may apply Liouville's theorem to the entire function gof
Exercise 3 Let f be an analytic function on D(0,1). Suppose that f(z) < 1 for all z € C and f() = 0. Show that G) . (Hint: use the function g(z) = f(2).)
2. Let f: R R be a continuous function. Suppose that f is differentiable on R\{0} and that there exists an L e R such that lim,of,(z) = L. Prove that f is differentiable at 1-0 with f,(0) = L. (Hint: Use the definition of derivative and then use mean value theorem) 2. Let f: R R be a continuous function. Suppose that f is differentiable on R\{0} and that there exists an L e R such that lim,of,(z) =...
Let Ω be an open set and a E Ω with (the closed disc) D(a,p) Ω Let f є H(Q). We have proved that for any r 〈 ρ, f has a power series expansion in the open disc D(a,r) CO 0 where, for all n0,1,2 7l Here C is the positively oriented circle: z-a+pe.θ, 0-θ-2π. In particular, f has a Taylor series expansion in D(a, r): f" (a) 2-a 0 This results in two consequences (will be shown in...
Let f: C→C be an entire, one-to-one function. (a) Explain why g()-f() f(0) is an entire 1-1 function (b) Explain why there exists0 such that B(O,e) C g(B(O, 1)). Hint: Open Mapping thm.] (c) Explain why Ig(z)2є if 221 . [Hint: g is 1-1.] (d) Since g(0)=0, g(z)=2h(z) for some entire function h(z). Explain why h(z) is never 0 (e) Show that there is a constant C>0 such that 1/h2)l C if21 (f) Deduce that 1/h (z) is a constant...
Let f: [0,1]→R be uniformly continuous, so that for every >0, there exists δ >0 such that |x−y|< δ=⇒|f(x)−f(y)|< for every x,y∈[0,1].The graph of f is the set G f={(x,f(x)) :x∈[0,1]}.Show that G f has measure zero Let f : [0, 1] → R be uniformly continuous, so that for every e > 0, there exists 8 >0 such that 2- y<83|f() - f(y)< € for every 1, 9 € [0,1]. The graph of f is the set Gj =...
·J (I) < 0 for all such y. (Hint: let g(x)--f(x) and use part (a)) 3. In this problem, we prove the Intermedinte Value Theorem. Let Intermediate Value Theorem. Let f : [a → R be continuous, and suppose f(a) < 0 and f(b) >0. Define S = {t E [a, b] : f(z) < 0 for allェE [a,t)) (a) Prove that s is nonempty and bounded above. Deduce that c= sup S exists, and that astst (b) Use Problem...
Suppose f is a continuous and differentiable function on [0,1] and f(0)= f(1). Let a E (0, 1). Suppose Vr,y(0,1) IF f'(x) 0 and f'(y) ±0 THEN f'(x) af'(y) Show that there is exactly f(ax) and f'(x) 0 such that f(x) one Hint: Suppose f(x) is a continuous function on [0, 1] and f(0) x € (0, 1) such that f(x) = f(ax) f(1). Let a e (0,1), there exists an Suppose f is a continuous and differentiable function on...
Complex Analysis: Suppose f(2) is an entire function and that there exists a real number Ro such that \f(2)] = 2l for any complex number z with [2] > Ro Prove that f is of the form f(x) = a + bz for all z E C.
1. Let U C IRt be open, UR be a function, a U and 0 A v E R" such that Dof(a) exists. Show that DAvf(a) exists for every 0 λ E R, and DAwf(a-λDuf(a). 3 marks 1. Let U C IRt be open, UR be a function, a U and 0 A v E R" such that Dof(a) exists. Show that DAvf(a) exists for every 0 λ E R, and DAwf(a-λDuf(a). 3 marks