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Let G be an open subset of C. Suppose f: GC is analytic except for isolated...
2. Let U be an open subset of R and let A be a compact subset of U. Suppose that f: U R is a iction of class C() aud...
2. Let U be an open subset of R and let A be a compact subset of U. Suppose that f: U R is a iction of class C() aud let F-(()e KIf(r, y) 0 and that Df does not vatish on E. Investigate whether Dis a Jordan region. annc
Theorem 2. Let E be an open subset of R² and suppose that fe C'(E). Let...
Theorem 2. Let E be an open subset of R² and suppose that fe C'(E). Let y(t) be a periodic solution of (1) of period T. Then the derivative of the Poincaré map P(8) along a straight line normal to r = {x E R x = y(t) - (0),O SE ST} at x = 0 is given by T P(0) = exp V. f(y(t)) dt. 4. Show that the system • = -y + (1 – 22 - y2)2...
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).)
Problem 5. Suppose that f: +C is analytic on an open set 12 containing the closed...
Problem 5. Suppose that f: +C is analytic on an open set 12 containing the closed half plane H = {2€ C: Im(x) > 0} and that there is a finite constant M with f() < M for all z H. 1. Show that da = f(i) x² +1 +00 2. Show that if o is a point in C with Im(a) > 0, then I (a) Im(a)' 22-2Re(a)x+ lajar (3) deduce sin (Bx) where 870
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...
5. Let Ω be open in C and consider the set U in Ω that has no limit points in Ω. For the sake of your imagination, 0 could be the set of isolated zeros or poles of some mero- morphic function. Let C...
5. Let Ω be open in C and consider the set U in Ω that has no limit points in Ω. For the sake of your imagination, 0 could be the set of isolated zeros or poles of some mero- morphic function. Let C be a simple closed curve in Ω\U oriented counter clockwise. Can there exist infinitely many points of U contained inside the region bounded by C? Explain
5. Let Ω be open in C and consider the...
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)...
*4, Let U be an open subset of R" and f:U-R" a function whose component functions have continuous partial derivatives. We say that f is an immersion if Dsf is injective for all v in U...
*4, Let U be an open subset of R" and f:U-R" a function whose component functions have continuous partial derivatives. We say that f is an immersion if Dsf is injective for all v in U and a submersion if Dof is surjective for allv in U. (a) Suppose that f:U-R" is an immersion. Prove that, for each v in U, we can find an open set V of U containing v, an open set W of R" containing f...
Let U be an open subset of R. Let f: U C Rn → Rm. (a) Prove that f is continuously differentiable if and only if for ea...
Let U be an open subset of R. Let f: U C Rn → Rm. (a) Prove that f is continuously differentiable if and only if for each a є U, for each E > 0, there exists δ > 0 such that for each x E U, if IIx-all < δ, then llDf(x)-Df(a) ll < ε. (b) Let m n. Prove that if f is continuously differentiable, a E U, and Df (a) is invertible, then there exists δ...
(6) Let a<b, and suppose the function f is integrable a, b. Show that for every...
(6) Let a<b, and suppose the function f is integrable a, b. Show that for every infinite on IR such that g(x)= f (x) for all e [a,b]\ S subset SC [a, b), there is a function g: [a, b and g is not integrable. [ef: 7.1.3 in text. (7) Show directly that if the function f : [a,b possibly at one point o (a,b), thenf is integrable on fa, b). R is continuous everywhere in a, b) except
(6)...
2. Let U be an open subset of R and let A be a compact subset of U. Suppose that f: U R is a iction of class C() aud let F-(()e KIf(r, y) 0 and that Df does not vatish on E. Investigate whether Dis a Jordan region. annc
Theorem 2. Let E be an open subset of R² and suppose that fe C'(E). Let y(t) be a periodic solution of (1) of period T. Then the derivative of the Poincaré map P(8) along a straight line normal to r = {x E R x = y(t) - (0),O SE ST} at x = 0 is given by T P(0) = exp V. f(y(t)) dt. 4. Show that the system • = -y + (1 – 22 - y2)2...
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).)
Problem 5. Suppose that f: +C is analytic on an open set 12 containing the closed half plane H = {2€ C: Im(x) > 0} and that there is a finite constant M with f() < M for all z H. 1. Show that da = f(i) x² +1 +00 2. Show that if o is a point in C with Im(a) > 0, then I (a) Im(a)' 22-2Re(a)x+ lajar (3) deduce sin (Bx) where 870
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...
5. Let Ω be open in C and consider the set U in Ω that has no limit points in Ω. For the sake of your imagination, 0 could be the set of isolated zeros or poles of some mero- morphic function. Let C be a simple closed curve in Ω\U oriented counter clockwise. Can there exist infinitely many points of U contained inside the region bounded by C? Explain
5. Let Ω be open in C and consider the...
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)...
*4, Let U be an open subset of R" and f:U-R" a function whose component functions have continuous partial derivatives. We say that f is an immersion if Dsf is injective for all v in U and a submersion if Dof is surjective for allv in U. (a) Suppose that f:U-R" is an immersion. Prove that, for each v in U, we can find an open set V of U containing v, an open set W of R" containing f...
Let U be an open subset of R. Let f: U C Rn → Rm. (a) Prove that f is continuously differentiable if and only if for each a є U, for each E > 0, there exists δ > 0 such that for each x E U, if IIx-all < δ, then llDf(x)-Df(a) ll < ε. (b) Let m n. Prove that if f is continuously differentiable, a E U, and Df (a) is invertible, then there exists δ...
(6) Let a<b, and suppose the function f is integrable a, b. Show that for every infinite on IR such that g(x)= f (x) for all e [a,b]\ S subset SC [a, b), there is a function g: [a, b and g is not integrable. [ef: 7.1.3 in text. (7) Show directly that if the function f : [a,b possibly at one point o (a,b), thenf is integrable on fa, b). R is continuous everywhere in a, b) except
(6)...
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