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Suppose f : B(0.1) C is holomorphic, with irg:) 1 for every z є B(0,1). Suppose also that f(0)-0, so f(z)g(2) for some holomorphic function g: B(0,1)C. (a) By applying the Maximum Principle to g on B...
[3] 5. Suppose that f: D[0,1] for all z E D[0, 1] D[0,1] is holomorphic, prove...
[3] 5. Suppose that f: D[0,1] for all z E D[0, 1] D[0,1] is holomorphic, prove that \f'(z) < 1/(1 - 121)2
Let f be a function which is holomorphic on the unit disk D (0). We define...
Let f be a function which is holomorphic on the unit disk D (0). We define the quantity R= sup \f (2) - f(w) ) ED (0) ( (which can be infinite). 1. Prove that Vr € (0,1), 2f'(0) = 2 1 f6)-f(-) .0) (2 where the circle C,(0) is traversed in the counterclockwise direction 2. Deduce from the previous result that 28' (O SR.
(3) 5. Suppose that f : D[0, 1] → D[0, 1] is holomorphic, prove that f'(2)...
(3) 5. Suppose that f : D[0, 1] → D[0, 1] is holomorphic, prove that f'(2) < 1/(1 - 121) for all z e D[0,1].
Suppose f(z) is a holomorphic function in a domain U, and z0 ∈ U. Prove that f has a zero of orde...
Suppose f(z) is a holomorphic function in a domain U, and z0 ∈ U. Prove that f has a zero of order m at z0 if and only if f(z) = g(z)(z − z0)^m, where g(z) is holomorphic in U and g(z0) not equal to 0. Please prove both directions of the if and only if statement and use series expansion to prove. We have not learned calculus of residues yet.
Exercise 3 Let f be an analytic function on D(0,1). Suppose that f(z) < 1 for...
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).)
Let f: [0,1]→R be uniformly continuous, so that for every >0, there exists δ >0 such...
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 =...
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 th...
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...
Problem 4. For r E [0, 1, fnd F)-(t)dt, where fr) 3 2r. Verify that F is continuous on [0,1] and F"(z) =f(z) a...
Problem 4. For r E [0, 1, fnd F)-(t)dt, where fr) 3 2r. Verify that F is continuous on [0,1] and F"(z) =f(z) at all points where f is continuous. Problern 5. Suppose that g, h : [c, d] → [a,b] are differentiable. ForエE [c,d] define h(a) Find H'(r)
Problem 4. For r E [0, 1, fnd F)-(t)dt, where fr) 3 2r. Verify that F is continuous on [0,1] and F"(z) =f(z) at all points where f is continuous. Problern...
9.) Suppose that X is a continuous random variable with density C(1- if r [0,1 0...
9.) Suppose that X is a continuous random variable with density C(1- if r [0,1 0 ¡f x < 0 or x > 1. (a) Find C so that px is a probability density function (b) Find the cumulative distribution of X (c) Calculate the probability that X є (0.1,0.9). (d) Calculate the mean and the variance of X 10.) Suppose that X is a continuous random variable with cumulative distribution function Fx()- arctan()+ (a) Find the probability density function...
[3] 5. Suppose that f: D[0,1] for all z E D[0, 1] D[0,1] is holomorphic, prove that \f'(z) < 1/(1 - 121)2
Let f be a function which is holomorphic on the unit disk D (0). We define the quantity R= sup \f (2) - f(w) ) ED (0) ( (which can be infinite). 1. Prove that Vr € (0,1), 2f'(0) = 2 1 f6)-f(-) .0) (2 where the circle C,(0) is traversed in the counterclockwise direction 2. Deduce from the previous result that 28' (O SR.
(3) 5. Suppose that f : D[0, 1] → D[0, 1] is holomorphic, prove that f'(2) < 1/(1 - 121) for all z e D[0,1].
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).)
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 =...
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...
Problem 4. For r E [0, 1, fnd F)-(t)dt, where fr) 3 2r. Verify that F is continuous on [0,1] and F"(z) =f(z) at all points where f is continuous. Problern 5. Suppose that g, h : [c, d] → [a,b] are differentiable. ForエE [c,d] define h(a) Find H'(r)
Problem 4. For r E [0, 1, fnd F)-(t)dt, where fr) 3 2r. Verify that F is continuous on [0,1] and F"(z) =f(z) at all points where f is continuous. Problern...
9.) Suppose that X is a continuous random variable with density C(1- if r [0,1 0 ¡f x < 0 or x > 1. (a) Find C so that px is a probability density function (b) Find the cumulative distribution of X (c) Calculate the probability that X є (0.1,0.9). (d) Calculate the mean and the variance of X 10.) Suppose that X is a continuous random variable with cumulative distribution function Fx()- arctan()+ (a) Find the probability density function...
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