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5.72. Let A = A(0,1) and let g: A → be an analytic function sat- isfying...
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).)
11. (8)(a) Suppose that f and g are analytic branches of zt on a domain D such that 0 g D Show th...
11. (8)(a) Suppose that f and g are analytic branches of zt on a domain D such that 0 g D Show that there is a fifth root, wo, of 1 such that f(z)-wog(2) for all E D. I suggest considering h(z) f (z)/g(z) (b) Now suppose that D D C(-,0]. Let f be an analytic branch of zt in D such that f (1) 1. Show that f(z) expLog(2)) for all z ED.
11. (8)(a) Suppose that f and...
= Q 6. Let f: {0,1} + {0,1} be a function given by f(0) = 0...
= Q 6. Let f: {0,1} + {0,1} be a function given by f(0) = 0 = f(1). Find two (total) functions g: {0,1} +{0,1} and h: {0,1} +{0,1} such that fog #gof and foh= hof. Write out f and your chosen functions g and h in table form, using a single table. Only the table is required as the answer.
13.1.11. Problem. Let f(x) = x and g(x) = 0 for all x ∈ [0,1]. Find...
13.1.11. Problem. Let f(x) = x and g(x) = 0 for all x ∈ [0,1].
Find a function h in B([0,1]) such that
du(f,h) = du(f,g) = du(g,h).
(3 problems)
13.2.6. Problem. Given in each of the following is the nth term of a sequence of real valued functions defined on (0, 1]. Which of these converge pointwise on (0, 1]? For which is the convergence uniform? (a) a z" (b) z+ nr. (c) a+ re-na 13.2.7. Problem. Given in...
(b) Let f 0, 1-R be a C2 function and let g, h: [0, 00)-R be C1. Consider the initial-boundary value problem kwr w(...
(b) Let f 0, 1-R be a C2 function and let g, h: [0, 00)-R be C1. Consider the initial-boundary value problem kwr w(r, 0) f(a) w(0, t) g(t) w(1, t) h(t) for a function w: [0,1 x [0, 0)- R such that w, wn, and wa exist and are continuous. Show that the solution to this problem is unique, that is, if w1 and w2 [0, 1] x [0, 00)- R both satisfy these conditions, then w1 = w2....
6. Let D be a bounded domain with boundary B. Suppose that f and g are both analytic on D and continuous on Du B, and suppose further that Re /(z)- Re g(z) for all z e B. Show that J-g + ία in D, whe...
6. Let D be a bounded domain with boundary B. Suppose that f and g are both analytic on D and continuous on Du B, and suppose further that Re /(z)- Re g(z) for all z e B. Show that J-g + ία in D, where α is a real constant.
6. Let D be a bounded domain with boundary B. Suppose that f and g are both analytic on D and continuous on Du B, and suppose further that...
(5) Let f: [0, 1 R. We say that f is Hölder continuous of order a e (0,1) if \f(x) -- f(y)| . , y sup [0, 1] with 2 # 1...
(5) Let f: [0, 1 R. We say that f is Hölder continuous of order a e (0,1) if \f(x) -- f(y)| . , y sup [0, 1] with 2 # 1£l\c° sup is finite. We define Co ((0, 1]) f: [0, 1] -R: f is Hölder continuous of order a}. = (a) For f,gE C ([0, 1]) define da(f,g) = ||f-9||c«. Prove that da is a well-defined metric Ca((0, 1) (b) Prove that (C ([0, 1]), da) is complete...
B2. (a) Let I denote the interval 0,1 and let C denote the space of continuous functions I-R. Def...
B2. (a) Let I denote the interval 0,1 and let C denote the space of continuous functions I-R. Define dsup(f,g)-sup |f(t)-g(t) and di(f.g)f (t)- g(t)ldt (f,g E C) tEI (i) Prove that dsup is a metric on C (ii) Prove that di is a metric on C. (You may use any standard properties of continuous functions and integrals, provided you make your reasoning clear.) 6 i) Let 1 denote the constant function on I with value 1. Give an explicit...
2. (a) Prove the product rule for complex functions. More specifically, if f(z) and g(z) f(z)g(z) is also analytic, and...
2. (a) Prove the product rule for complex functions. More specifically, if f(z) and g(z) f(z)g(z) is also analytic, and that analytic prove are that h(z) h'(z)f(z)9() f(z)g'(z) (You may use results from the multivariable part of the course without proof.) = nz"- for n e N = {1,2,3,...}. Your textbook establishes that S z"= dz (b) Let Sn be the statement is true. With the help of (a), show that if Sn is true, then Sn+1 is true. Why...
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...
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(0, r) where 0 < r < 1 , deduce that If( S for every 2E (0, 1) . (b) Show also that |f'(0) S1 (c) Show that if lf(z)- for some z B(0,1)\(0), or if If,(0)| = 1 , then there is a...
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).)
11. (8)(a) Suppose that f and g are analytic branches of zt on a domain D such that 0 g D Show that there is a fifth root, wo, of 1 such that f(z)-wog(2) for all E D. I suggest considering h(z) f (z)/g(z) (b) Now suppose that D D C(-,0]. Let f be an analytic branch of zt in D such that f (1) 1. Show that f(z) expLog(2)) for all z ED.
11. (8)(a) Suppose that f and...
= Q 6. Let f: {0,1} + {0,1} be a function given by f(0) = 0 = f(1). Find two (total) functions g: {0,1} +{0,1} and h: {0,1} +{0,1} such that fog #gof and foh= hof. Write out f and your chosen functions g and h in table form, using a single table. Only the table is required as the answer.
13.1.11. Problem. Let f(x) = x and g(x) = 0 for all x ∈ [0,1].
Find a function h in B([0,1]) such that
du(f,h) = du(f,g) = du(g,h).
(3 problems)
13.2.6. Problem. Given in each of the following is the nth term of a sequence of real valued functions defined on (0, 1]. Which of these converge pointwise on (0, 1]? For which is the convergence uniform? (a) a z" (b) z+ nr. (c) a+ re-na 13.2.7. Problem. Given in...
(b) Let f 0, 1-R be a C2 function and let g, h: [0, 00)-R be C1. Consider the initial-boundary value problem kwr w(r, 0) f(a) w(0, t) g(t) w(1, t) h(t) for a function w: [0,1 x [0, 0)- R such that w, wn, and wa exist and are continuous. Show that the solution to this problem is unique, that is, if w1 and w2 [0, 1] x [0, 00)- R both satisfy these conditions, then w1 = w2....
6. Let D be a bounded domain with boundary B. Suppose that f and g are both analytic on D and continuous on Du B, and suppose further that Re /(z)- Re g(z) for all z e B. Show that J-g + ία in D, where α is a real constant.
6. Let D be a bounded domain with boundary B. Suppose that f and g are both analytic on D and continuous on Du B, and suppose further that...
(5) Let f: [0, 1 R. We say that f is Hölder continuous of order a e (0,1) if \f(x) -- f(y)| . , y sup [0, 1] with 2 # 1£l\c° sup is finite. We define Co ((0, 1]) f: [0, 1] -R: f is Hölder continuous of order a}. = (a) For f,gE C ([0, 1]) define da(f,g) = ||f-9||c«. Prove that da is a well-defined metric Ca((0, 1) (b) Prove that (C ([0, 1]), da) is complete...
B2. (a) Let I denote the interval 0,1 and let C denote the space of continuous functions I-R. Define dsup(f,g)-sup |f(t)-g(t) and di(f.g)f (t)- g(t)ldt (f,g E C) tEI (i) Prove that dsup is a metric on C (ii) Prove that di is a metric on C. (You may use any standard properties of continuous functions and integrals, provided you make your reasoning clear.) 6 i) Let 1 denote the constant function on I with value 1. Give an explicit...
2. (a) Prove the product rule for complex functions. More specifically, if f(z) and g(z) f(z)g(z) is also analytic, and that analytic prove are that h(z) h'(z)f(z)9() f(z)g'(z) (You may use results from the multivariable part of the course without proof.) = nz"- for n e N = {1,2,3,...}. Your textbook establishes that S z"= dz (b) Let Sn be the statement is true. With the help of (a), show that if Sn is true, then Sn+1 is true. Why...
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(0, r) where 0 < r < 1 , deduce that If( S for every 2E (0, 1) . (b) Show also that |f'(0) S1 (c) Show that if lf(z)- for some z B(0,1)\(0), or if If,(0)| = 1 , then there is a...
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