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11. (8)(a) Suppose that f and g are analytic branches of zt on a domain D such that 0 g D Show th...
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...
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).)
12. Suppose that fis analytic on a convex domain D and that Re(f ,(z)) > 0 for all z E D. Show that f is one-to-one on D. (Hint: /(z2) - sz) J,f'(w) dw, where is the line segment joining z1 to...
12. Suppose that fis analytic on a convex domain D and that Re(f ,(z)) > 0 for all z E D. Show that f is one-to-one on D. (Hint: /(z2) - sz) J,f'(w) dw, where is the line segment joining z1 to z2.)
12. Suppose that fis analytic on a convex domain D and that Re(f ,(z)) > 0 for all z E D. Show that f is one-to-one on D. (Hint: /(z2) - sz) J,f'(w) dw, where is the...
(b) Suppose f is analytic in a neighborhood of the closed and bounded nonempty domain D....
(b) Suppose f is analytic in a neighborhood of the closed and bounded nonempty domain D. Assuming that f(z) # 0 for all z e D, show that the minimum value of \f| is achieved on the boundary ƏD of D. (Bonus: can you give a contradiction to the above statement if you assume f vanishes at a point in the interior of D?)
4[10 pts]. Let f(z) = u (r,0) + iv(r,0) be analytic in a domain D c C which does not contain the ...
4[10 pts]. Let f(z) = u (r,0) + iv(r,0) be analytic in a domain D c C which does not contain the origin. Then do the following ones: (a) Show that rurr(r, θ) + rur(r, θ) + u69(r, θ) 0 for all re® E D. (b) Show that (a) is equivalent to the condition that u is harmonic in D (c) Show that the function (in|e )2-[Arg( a(z) z)]2,-π < Arg(z) < π,
4[10 pts]. Let f(z) = u (r,0)...
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)...
8. Let f be analytic on a bounded domain D and continuous on Du B, where B is the boundary of D. ...
8. Let f be analytic on a bounded domain D and continuous on Du B, where B is the boundary of D. Show that if f is never zero on D, then the minimum of lfl is assumed on B. You will need to use the fact that lfl does, indeed, assume a minimum somewhere on D B
8. Let f be analytic on a bounded domain D and continuous on Du B, where B is the boundary of D....
Please answer correctly. that f tC2) s uestion Show analytic and real-valued on a- domain D)...
Please answer correctly.
that f tC2) s uestion Show analytic and real-valued on a- domain D) then t(2) is constant
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...
u(20) for all z e D. Prove tha E C:0<zl<2) and Cr be the positively oriented 9 (10) Suppose that f is analytic in the deleted disk B2(0) C be the positi that If(2)l S M<oo for all z e B2...
u(20) for all z e D. Prove tha E C:0<zl<2) and Cr be the positively oriented 9 (10) Suppose that f is analytic in the deleted disk B2(0) C be the positi that If(2)l S M<oo for all z e B2(0). If 0 TS circle |zl r. Show that S 1, then let Cr r | 1= f(z) dz = 0. (Hint: why is the value of (1) the same if C, is replaced by C?
u(20) for all z...
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...
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).)
12. Suppose that fis analytic on a convex domain D and that Re(f ,(z)) > 0 for all z E D. Show that f is one-to-one on D. (Hint: /(z2) - sz) J,f'(w) dw, where is the line segment joining z1 to z2.)
12. Suppose that fis analytic on a convex domain D and that Re(f ,(z)) > 0 for all z E D. Show that f is one-to-one on D. (Hint: /(z2) - sz) J,f'(w) dw, where is the...
(b) Suppose f is analytic in a neighborhood of the closed and bounded nonempty domain D. Assuming that f(z) # 0 for all z e D, show that the minimum value of \f| is achieved on the boundary ƏD of D. (Bonus: can you give a contradiction to the above statement if you assume f vanishes at a point in the interior of D?)
4[10 pts]. Let f(z) = u (r,0) + iv(r,0) be analytic in a domain D c C which does not contain the origin. Then do the following ones: (a) Show that rurr(r, θ) + rur(r, θ) + u69(r, θ) 0 for all re® E D. (b) Show that (a) is equivalent to the condition that u is harmonic in D (c) Show that the function (in|e )2-[Arg( a(z) z)]2,-π < Arg(z) < π,
4[10 pts]. Let f(z) = u (r,0)...
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)...
8. Let f be analytic on a bounded domain D and continuous on Du B, where B is the boundary of D. Show that if f is never zero on D, then the minimum of lfl is assumed on B. You will need to use the fact that lfl does, indeed, assume a minimum somewhere on D B
8. Let f be analytic on a bounded domain D and continuous on Du B, where B is the boundary of D....
Please answer correctly.
that f tC2) s uestion Show analytic and real-valued on a- domain D) then t(2) is constant
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...
u(20) for all z e D. Prove tha E C:0<zl<2) and Cr be the positively oriented 9 (10) Suppose that f is analytic in the deleted disk B2(0) C be the positi that If(2)l S M<oo for all z e B2(0). If 0 TS circle |zl r. Show that S 1, then let Cr r | 1= f(z) dz = 0. (Hint: why is the value of (1) the same if C, is replaced by C?
u(20) for all z...
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