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Show that the function defined by (2)= is analytic on the open unit disk {lz| <...
Exercise 6.7. Show that the space of functions analytic in the open unit disk and such that (6.8)...
Exercise 6.7. Show that the space of functions analytic in the open unit disk and such that (6.8) is a Banach space with the norm (6.8) (part of the exercise is to show that the latter indeed zED defines a norm).
Exercise 6.7. Show that the space of functions analytic in the open unit disk and such that (6.8) is a Banach space with the norm (6.8) (part of the exercise is to show that the latter indeed zED defines...
5.72. Let A = A(0,1) and let g: A → be an analytic function sat- isfying...
5.72. Let A = A(0,1) and let g: A → be an analytic function sat- isfying 9(0) = 0 and 1g'(0) = 1 whose derivative is a bounded function in A. Show that w > (4m)-1 for every point w of C ~ g(A), where m = sup{]g'(x): z E A}; i.e., show that the range of g contains the disk A(0,(4m)–?). (Hint. Fix w belonging to C ~ g(A). Then w # 0. The function h defined by h(z)...
(i) Give an example of a function f(x,y) that is defined and continuous on the closed unit disk B...
(i) Give an example of a function f(x,y) that is defined and continuous on the closed unit disk B(0) ((,y) E R2 but does not achieve a maximum on the punc- 2 marks] tured closed disk B.(0 )"-{ (z, y) E R2 10c x2 + y2 < 1}
(i) Give an example of a function f(x,y) that is defined and continuous on the closed unit disk B(0) ((,y) E R2 but does not achieve a maximum on the punc- 2...
Q1. Show analytically that the Root Locus for the unity feedback system with open loop transfer f...
Q1. Show analytically that the Root Locus for the unity feedback system with open loop transfer function: (a) [10 marks] K(s 4) (s + 2) is a circle, and find the centre and the radius. Determine the minimum value of the damping ratio and the corresponding value of K (b) The root locus of the open loop transfer function: [10 marks] s(s26s +15) is depicted in Figure Q1(b). Find the minimum value of gain K that will render the system...
Suppose is some sequence of holomorphic functions, which are defined on an open set containing the...
Suppose
is some sequence of holomorphic functions, which are defined on an
open set containing the closed unit disk
.
Suppose also that
converges uniformly on the unit circle
.
Show then that
converges to a holomorphic function
on
9n We were unable to transcribe this image9n aD 9n We were unable to transcribe this imageWe were unable to transcribe this image
55. Show that a monotone function on an open interval is continuous if and only if...
55. Show that a monotone function on an open interval is continuous if and only if its image is an interval. 56. Let f be a real-valued function defined on R. Show that the set of points at which f is continuous is a Gs set.
12.4. The Bessel function (of the first kind of order ser is defined by Jo(+) 2...
12.4. The Bessel function (of the first kind of order ser is defined by Jo(+) 2 (nl) This function is of considerable importance in applied mathematics, with merous applications to problems involving cylindrical containers, such as temperature distribution in a se pipe (6) Write out the partial num for the Bessel function of order sero up to the terms. If you have access to a graphing calculator or computer, use the graph of this partial sum to approximate, to one...
Problem 5. (20pts) The open-loop transfer function of a unity feedback system G(8) -- +2) a)...
Problem 5. (20pts) The open-loop transfer function of a unity feedback system G(8) -- +2) a) Locate open-loop zeros and open-loop poles. b) Construct the root-locus diagram as 0 <K <oo. Mark the portions of the real axis that belong to the root locus - Mark with K =0 the point where the root locus bra O the point where the root locus branches start and with K = oo the point where the branches end. - Find break-away and/or...
Problem (3) A function f(z) is analytic in the disk -1 where the modulus satisfies the bound Here b 2 a > 0 Find an...
Problem (3) A function f(z) is analytic in the disk -1 where the modulus satisfies the bound Here b 2 a > 0 Find an optimal bound on |f'(0) in terms of a and b. Complete arguments required By optimal it is meant that (1) the bound holds for all functions with the stated property and (2) there actually is a function with the stated property such that the bound holds as an equality. The second part of this problem...
Q5. a) Let f(z) be an analytic function on a connected open set D. If there...
Q5. a) Let f(z) be an analytic function on a connected open set D. If there are two constants and C, EC, not all zero, such that cf(z)+ f(2)=0 for all z € D, then show that f(z) is [4] a constant on D. b) Evaluate the contour integral f(z)dz using the parametric representations for C, where f(2)= and the curve C is the right hand half circle 1z| = 2, from z=-2 to z=2i. [4] c) Evaluate the contour...
Exercise 6.7. Show that the space of functions analytic in the open unit disk and such that (6.8) is a Banach space with the norm (6.8) (part of the exercise is to show that the latter indeed zED defines a norm).
Exercise 6.7. Show that the space of functions analytic in the open unit disk and such that (6.8) is a Banach space with the norm (6.8) (part of the exercise is to show that the latter indeed zED defines...
5.72. Let A = A(0,1) and let g: A → be an analytic function sat- isfying 9(0) = 0 and 1g'(0) = 1 whose derivative is a bounded function in A. Show that w > (4m)-1 for every point w of C ~ g(A), where m = sup{]g'(x): z E A}; i.e., show that the range of g contains the disk A(0,(4m)–?). (Hint. Fix w belonging to C ~ g(A). Then w # 0. The function h defined by h(z)...
(i) Give an example of a function f(x,y) that is defined and continuous on the closed unit disk B(0) ((,y) E R2 but does not achieve a maximum on the punc- 2 marks] tured closed disk B.(0 )"-{ (z, y) E R2 10c x2 + y2 < 1}
(i) Give an example of a function f(x,y) that is defined and continuous on the closed unit disk B(0) ((,y) E R2 but does not achieve a maximum on the punc- 2...
Q1. Show analytically that the Root Locus for the unity feedback system with open loop transfer function: (a) [10 marks] K(s 4) (s + 2) is a circle, and find the centre and the radius. Determine the minimum value of the damping ratio and the corresponding value of K (b) The root locus of the open loop transfer function: [10 marks] s(s26s +15) is depicted in Figure Q1(b). Find the minimum value of gain K that will render the system...
Suppose
is some sequence of holomorphic functions, which are defined on an
open set containing the closed unit disk
.
Suppose also that
converges uniformly on the unit circle
.
Show then that
converges to a holomorphic function
on
9n We were unable to transcribe this image9n aD 9n We were unable to transcribe this imageWe were unable to transcribe this image
55. Show that a monotone function on an open interval is continuous if and only if its image is an interval. 56. Let f be a real-valued function defined on R. Show that the set of points at which f is continuous is a Gs set.
12.4. The Bessel function (of the first kind of order ser is defined by Jo(+) 2 (nl) This function is of considerable importance in applied mathematics, with merous applications to problems involving cylindrical containers, such as temperature distribution in a se pipe (6) Write out the partial num for the Bessel function of order sero up to the terms. If you have access to a graphing calculator or computer, use the graph of this partial sum to approximate, to one...
Problem 5. (20pts) The open-loop transfer function of a unity feedback system G(8) -- +2) a) Locate open-loop zeros and open-loop poles. b) Construct the root-locus diagram as 0 <K <oo. Mark the portions of the real axis that belong to the root locus - Mark with K =0 the point where the root locus bra O the point where the root locus branches start and with K = oo the point where the branches end. - Find break-away and/or...
Problem (3) A function f(z) is analytic in the disk -1 where the modulus satisfies the bound Here b 2 a > 0 Find an optimal bound on |f'(0) in terms of a and b. Complete arguments required By optimal it is meant that (1) the bound holds for all functions with the stated property and (2) there actually is a function with the stated property such that the bound holds as an equality. The second part of this problem...
Q5. a) Let f(z) be an analytic function on a connected open set D. If there are two constants and C, EC, not all zero, such that cf(z)+ f(2)=0 for all z € D, then show that f(z) is [4] a constant on D. b) Evaluate the contour integral f(z)dz using the parametric representations for C, where f(2)= and the curve C is the right hand half circle 1z| = 2, from z=-2 to z=2i. [4] c) Evaluate the contour...
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